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The Holy Grail of MHP

Here is a short and elementary and complete solution of the MHP, which actually covers the biased host situation just as well as the usual symmetric case. There is no computation of a conditional probability. All we have to do is to consider two kinds of players: a player who in some situations would stay, and a player who in all situations would switch. We show that both kinds of players are going to end up with a goat with probability at least 1/3. In other words, it's not possible to do better than to get the car with probabillity 2/3. But always switching does give you the car with probability 2/3. Hence always switching achieves the best that you can possibly do.

Suppose all doors are equally likely to hide the car, and you choose Door 1.

If you are planning to stick to Door 1 if offered the choice to switch to Door 2, you'll not get the car if it is behind Door 2. In that case Monty would certainly open Door 3, you'll have the choice between Doors 1 and 2, and you'll keep to Door 1. Chance 1 in 3.

Similarly if you are planning to stick to Door 1 if offered the choice to switch to Door 3, you'll not get the car if it is behind Door 3. Probability 1/3.

If on the other hand you are planning to switch anyway, you'll not get the car if it is behind Door 1. Chance 1 in 3.

Altogether this covers every possible way of playing, and however Monty chooses his door: there's always a chance of at least 1/3 that you'll end up with a goat. This means that there is no way you can do better than getting the car with chance 2/3.

We know that "always switching" guarantees you *exactly* a chance of 2/3 of getting the car. I've just shown you that there is no way this can be improved.

Side remark 1: For those who are interested in conditional probabilities, the previous remarks prove that the conditional probabilities of the location of the car (given you chose door x and the host opened door y) will always be in support of switching. Otherwise, we could improve on the 2/3 overall succcess-chance of always switching, by not switching in a situation indicated by the conditional probability of winning by switching being less than 1/2.

Side remark 2: For those who are worried that I did not talk about randomized strategies (e.g. you toss an unbiased coin to decide whether to switch or stay, when you chose Door 1 and the host opened Door 3) it suffices to remark that you could as well have tossed your coin in advance of the host opening a door. Thus this is the same as choosing a deterministic strategy in advance, by randomization. Since any deterministic strategy gives you a goat with probability at least 1/3, the same is true when you choose one such strategy at random.

Of course 20 text-books in elementary probability theory do MHP in a different way, while the previous analysis is only implicit in the recent papers of A.V. Gnedin on arXiv.org, and on my home page [1]. Which might or might not be considered a "reliable source".Richard Gill (talk) 02:02, 2 July 2011 (UTC)

@ Gill110951 The fact that 2/3 is unbeatable for arbitrary q's is found in Morgan et al

"We see that one cannot do better than 2/3 in the unconditional game, and that the vos Savant scenario maximizes the overall efficacy of the switch strategy"

This can be shown combinatorially, I mentioned it before, here is a repetition. Let Conie plays strategy A. Then for some u(A) door u never brings her luck, whichever Monte's policy of revealing the doors. This is stronger than probability statement you make, because the unlucky door is universally unlucky, while it would be enough to have u depending on Monty's strategy as well. I read the complains of previous editors, and at the moment I am sympathetic with conditionalists. Their concern can be answered in von Mises style, without assumption of the ***existence*** of conditional probability as a long-run frequency. Machtindex (talk) 22:32, 1 July 2011 (UTC)

@Machtindex: I know that Morgan et al. also proved 2/3 is unbeatable. My "holy grail" was to find a one line verbal proof which will convince ordinary folk. Now you say the proof above is probabilistic not combinatorial. But I think it is combinatorial and it is the same as your proof. I showed for each deterministic strategy A how to determine a u(A). It's not quite a one-liner, but nearly. However you play there's always a door such that if the car is there, you'll miss it. We only need consider two cases: for "always switching" u(A) is the door you initially chose, and for "sometimes switching" u(A) is a door you wouldn't switch to if you had the option. Ordinary readers won't be interested in randomized strategies but anyone who wants to include these will understand how to do it. Now u(A) has to be a random door, depending on the same coin tosses used to implement A, hence still independent of the location of the car and hence hiding the car with probability 1/3.(By the way, shouldn't Conie be Connie? The girl's name? Richard Gill (talk) 01:30, 2 July 2011 (UTC)

I do not take sides in the simplists versus conditionalists arguments, but I am looking for easy bridges between the two points of view, with the hope of defusing the conflict. The closer the two kinds of solutions are brought together, the less the article has to be "hung up" about the distinction. But maybe other wikipedia editors don't find the new arguments attractive. And since they are not (yet) prominently present in the existing literature every editor has a powerful argument to ignore them. Richard Gill (talk) 01:43, 2 July 2011 (UTC)

@Gill110951

Origin of Conie The Name Conie is a girl's name . The origin of the baby name Conie is Latin with the meaning(s) depending on Gender/Origin being Latin- Constant, Steadfast. A short form of Constance. Conie has the following similar or variant Names: Conee Coney Coni Conie Connee conney Conni Conny Cony Connie Conie Name Popularity The name Conie, is the 35835th most popular baby name at mybaby.net.au placing it in the top 48% of names on our site.Machtindex (talk) 06:59, 2 July 2011 (UTC)

Conditional probability need not exist, but the Law of Large Numbers is partially capable of resolving The Great Big-Endian / Little-Endian Controversy

@ Rick Block.

Citation:

The probability the car is behind door 2 after the host opens door 3 is (by the elementary definition of conditional probability - no Bayes theorem required) the probability the car is behind door 2 divided by the sum of this plus the probability the car is behind door 1 (in the case the host opens door 3). If at the start the door 1 and door 2 probabilities are the same, then the resulting probability can obviously never be less than 1/2 - and if we assume the host picks evenly in the case the car is behind door 1, with very little thought, can be shown to be exactly 2/3 (1/3 / (1/3 + 1/6)). -- Rick Block (talk) 15:37, 29 June 2011 (UTC)

Dear Rick Block,

the question about the famous situation is very difficult to answer, as I see it now, because it depends on what Conie wants in the long run, which consequences on future rounds her decision will have and how Monte's future behavior will depend on this decision. One problem, and this is a minor one, is that the conditional probability may be undefined. So let us stick for a while with Conie's goal to win possibly high proportion of famous situations in n rounds for large n.

So let us start with a simple version. Suppose Conie switches in the famous situation. n rounds total, with switching from the famous situation bring wins about n/3 times, because the prize behind D2 is hidden some S times, and the law of large numbers says that S/n is close to 1/3. The prize is behind D1 and switch to D2 is offered some R rounds, in these cases Conie is also in the famous situation. This R is not larger than T, where T is the number of rounds the prize is behind D1. Therefore the proportion of games won in the famous situation with switching among the number of occurences of the famous situation is Q:= S/(S+R) which more than or equal to S/(S+T) because R < or = T. By the law of large numbers S/(S+T) is close to 1/2, (the average of the ratio is exactly 1/2). Therefore Q for large n is at least 1/2. This is all what Conie needs to convince herself that the gambling with switching is favorable for her in the sense of the proportion of the long-run wins in the famous situation.

The same, albeit with more details: The problem is not understanding the definition of P(A|B) as in your remark, the problem is deeper -- it is just the interpretation of probability as long run limiting proportion.

Let B be the event that the prize is behind D1. It has probability 1/3, meaning that out of n trials, the number T of trials "the prize is at D1" is about n/3, so T/n is close to 1/3. This is the very meaning of objective probability.

Out of these T cases Monte chooses some R cases to reveal door3. Now, while T/n becomes almost constant for large n, by the law of large numbers, the ratio R/T can in principle do what Monte wants. The limit of this ratio, if the limit makes sense at all, is your lovely q=1/2, is not it? If Monte uses random device -- fair coin -- to make his choice when he can independently (!!!) for all corresponding rounds, then q perfectly makes sense, and the definition of conditional probability is fine, because the famous situation will appear about n(1+q)/3 times.

But when R/T behaves like f(n)=(sin(n)+1)/2 - I hope the sine function is still not a rocket science for 15 yrs old, and I hope everybody understands that n can be plugged for "x", then the conditional probability is in a big trouble. Why? When you change n to n+1 then the value of sine jumps dramatically, so the limit R/T does not exist. To be very precise, take R= (integer part of n*f(n)), which is the closest integer from below to what is in brackets. Still no rocket science.

However if S is the number of times out of n when the prize is behind D2, switching wins in S cases, which is S/(S+R) of the time. This is of course, a surrogate finite forma of conditional probability, with Laplacian definition. I count the number of trials of interest S+R (famous situation) and the number of wins with switching S. Now, the law of large numbers tells me that S/n is about 1/3, and I know that R is not larger than T. From this every 15 yrs old, not very bright, is expected to conclude that Q= S/(S+R) is at least r = S/(S+T). With the little idea of symmetry 1=S/(S+T)+ T/(S+T) you see that in the mean (on the average) E r = 1/2 , and for large n, r is about 1/2 as the proportion of cases when the prize is behind D2 among the cases it is behind union D1 and D2). Thus in the long run Q is at least 1/2. Conie wins at least half of famous situations, when switching in the famous situations.

The role of your q is taken by the proportion of trials when Conie wins with switching to D2 (D1 is always chosen) among the trials that you see the famous situation. I am not very precise about "large n". But this is what the probability theory is about. It is the law of large numbers.

With switching in the famous situation Conie wins at least half of the cases when the famous situation occurs. If she is bounded to play in only this situation, leaving the play in the case D1-chosen-D2-revealed to her boyfriend, then *at least* for about a half of the repetitions of the famous situation she will win. If she gambles in the famous situations only, winning just a cent when the final guess is OK, she will switch to earn her living every time. And she will pay all her loans by earning cent-by-cent if R/T will not exceed 99% in the long run, provided the "long" is long enough. We could be sure that The Reverend Thomas Bayes would had come to my conclusion.

So far so good. But now we see that the gambler may have distinct goals.

Total: maximize long-run proportion of wins in n games

Restricted: maximize the long-run proportion of wins in famous situations.

FiniteRestricted: For finite fixed n, also Maximize number of famous situations won among n first rounds

FiniteTotal: maximize wins in n games

The word "maximize" subtends something yet unclear about Monte's behavior, which co-determines the number of wins. Monte's behavior can be history-dependent, and this makes things difficult.

Suppose Conie wants FiniteTotal and plays always switch. Let she wins cent when guesses right, and loses otherwise, then her capital is a submartingale, no matter what history-dependent Monte does. The famous situations occur at stopping times, hence Doobs optional sampling tells us that the capital process restricted to famous situations is a submartingale. Hence switching in famous situations is favorable.

The same for Total criterion.

If the goal is FiniteRestricted then switching maybe bad. Suppose Monte's reaction on the first switching is creating less famous situations in the future, within n rounds. Then Conie plays rarely and wins less in n rounds.

If the goal is FiniteTotal I do not yet know know what to say.

Summary: the switching dilemma in famous situation must be associated with some finite-n or asymptotic criterion of the capital won. If lovely q is something coming from independent tosses of a coin, then long-run winning proportion with switching in the famous situation is well defined. For other Monte's policies maybe not.

Formulation of the dilemma about the probability requires some frequency of wins interpretation, but the series of trials is co-determined by Monte who does not guarantee playing to produce some frequencies of wins. Naive approaches like "what is the probability to win in famous situation with switching" may fail. — Preceding unsigned comment added by Machtindex (talk • contribs) 22:11, 2 July 2011 (UTC)

Is this analysis from some source, or is it your own? It mostly sounds like a game theoretic sort of approach (which is definitely underrepresented in the current article). In the quote above, I was talking about the "usual" problem, where the car is uniformly distributed at the beginning and we then assume the host uniformly picks which door to open if he has a choice (in which case the probability of winning by switching is exactly 2/3). There are many "unusual" variations - the main bone of contention here (AFAICT) seems to be whether a solution based on conditional probability would be useful to most readers and should be presented for the usual problem on a par with the "simple" solutions (which show a 2/3 chance of winning for a strategy of switching, as distinct from a 2/3 chance of winning given the player has picked door 1 and has seen the host open door 3), or whether the machinery of conditional probability is inherently beyond the comprehension of nearly all readers and, thus, should be confined to a later section of the article intended only for experts. It is my firm belief that we should show both "simple" and conditional solutions and that placing one in a position of primacy to the other creates a bias, which the article must not have (per wp:npov). Both of these kinds of solutions are extremely common in reliable sources. To fairly represent the sources, we must not favor one over the other. -- Rick Block (talk) 00:51, 3 July 2011 (UTC)

@ Rick Block, sure there is a source -- my brains.

I agree that both conditional and unconditional forms of the question are important. But I feel that the exact Bayesian computation is not so sound, and its explanation is not good for laymen. The laymen thinks in terms of proportions, not intersections of abstract event.

Where surprise is coming from: we are first under conditional illusion 50:50, but unconditional constant strategy tells us this is wrong. Quantification of the "famous situation" is a highly ambiguous issue because it involves powerful informed player, which by very strange reasons is switched off by a constant policy. Probably, Markov decision processes can suitably address the host's behavior.

Let us pass to more constructive discussion. An obvious (to me) thing to start with is to organise game-theoretic section. If you are concerned about published/unpublished, consult

Ole Haggstrom's book (Streifzuege, 

German edition 2005, translated from Swedish). I hope 7 years is enough, to consider the theory established. I can solicit help of good people, given guarantee that I am not wasting my effort.

>>>>>To fairly represent the sources, we must not favor one over the other.

I am not so sure. Most of the sources are awkward, and those which are reliable copy one another, thus in essense the site represents very limited number of basic sources, which interested people have seen elsewhere. The Admin should have taste to math, to filter the sources.93.206.174.81 (talk) 22:41, 3 July 2011 (UTC)

Can we agree on what to name each position?

OK, I am going to do a simple experiment. As you know, I am of the opinion that the fundamental content dispute here should be walked through the steps of content dispute resolution, and to start that process I need something short that describes the two positions. To that end, I am once again asking for help, this time on one small part of the puzzle:

Can we agree on what to name each position? Hopefully in plain English that is non-prejudicial, unbiased and descriptive?

If we cannot agree on even that much, I am wasting my time here. Guy Macon (talk) 19:42, 3 July 2011 (UTC)

Guy Macon, I think the positions become diffuse. I understand there are several social conflicts here.

The most frustrating for me is that excellent minds left this forum as they were not given space to develop their scientific and, what is even worth, didactical ideas. Richard Gill mentioned that he had difficulties to put on the page arguments for the fundamental upper bound 2/3: this is a shame for administrators. Grisha Perelman did not publish his famous papers at all just posted on the arxiv.org -- following the MHP-admins policy the solution to Poincare conjecture remains unpublished, thus not worth mentioning on Wiki.

Another conflict is for the "right solution" and the weights of solutions to be presented. Some more interesting and fundamental issue feeding the big/little-endian quarrel is finding good bridges from unconditional to conditional.

The site seems to me good in the main part, but it is dead because interesting things for possibly student's course works are not here, and one needs to look for more competent sources.

Basically, the always-switching strategy is so robust, that Marylyn will be always right laughing at such discussions. But we are researchers, and we should explore various formulations, formalisations and solutions. For me personally the question about probabilities is of the second order. It is only one possible quantification of strategies.

To show you another option: here it is. Play 3000 times, some A,B,C of 3000 times the prize is at D1,D2,D3.

The famous situation occurs R+B times, where R number of times of D1 the switch to D2 offered.

I win B times out of R+B, and E(B-R)>=E(B-A)=0 by symmetry. Thus on the average switching from famous situation gives more wins that staying. Mathematically, we do not need even independence of trials, nor condition probs which may stay undefined. Why cannot such elementary things appear on the page, upon approval of the discussants? Or should such highschool-student-level observation wait publication in Inventiones?

That is the point. Bright findings should be presented to inspire genuinely interested people to come here. The Admin should have taste for interesting things, like journal editor.93.206.174.81 (talk) 22:41, 3 July 2011 (UTC)

There are no admins in the sense you are using - nobody with even the slightest hint of any special authority over content. Wikipedia simply does not work that way. The founder and head of Wikipedia is the same as a 10-year-old editing Wikipedia for the first time using an IP address (not using a username).

There are editors, though, and you are one of them! You, personally, can present bright findings and encourage others to do so as well.

As for "following the MHP-admins policy X remains unpublished, thus not worth mentioning on Wiki" here is a quote that sums up my feelings on that:

"If Wikipedia had been available around the fourth century B.C., it would have reported the view that the Earth is flat as a fact and without qualification. And it would have reported the views of Eratosthenes (who correctly determined the earth's circumference in 240BC) either as controversial, or a fringe view. Similarly if available in Galileo's time, it would have reported the view that the sun goes round the earth as a fact, and Galileo's view would have been rejected as 'original research'. Of course, if there is a popularly held or notable view that the earth is flat, Wikipedia reports this view. But it does not report it as true. It reports only on what its adherents believe, the history of the view, and its notable or prominent adherents. Wikipedia is inherently a non-innovative reference work: it stifles creativity and free-thought. Which is A Good Thing." --WP:FLAT

I would also like to point out that various other encyclopedias have been put online that differ from Wikipedia in key policies that are often criticized here. All have failed, despite the face that Wikipedia will give you - at no charge - a copy of the Wikipedia software and all of the existing Wikipedia articles, just to make it easy to start your own encyclopedia.

So, anyone want to report any progress on naming each position? --Guy Macon (talk) 02:37, 4 July 2011 (UTC)

Guy Macon, I had seen the passage about the flat Earth. There is however much difference. I understand the MHP article is mainly about mathematics, which is a collection of rules and logical statements. Everybody of editors is able to check bright didactical findings, which w.r.t complexity do not go beyond high school level btw.Machtindex (talk) 14:00, 4 July 2011 (UTC)

We need "something short that describes the two positions"

I propose the following description:

A) Observers of door tickets: door watchers – The article is not about finding the answer  switch or stay,  but about the correct technology in using Bayes and about conditioning on door numbers, although completely irrelevant to correctly answer the famous question (being a "door watcher").

B) Yes switch now: universalists grant 2/3 – The article should clearly lay out to grandma and grandson why switching can never be beaten by any other decision, as always switching guarantees 2/3 and no other method can ever do better (being a "universalist").

Please will s.o. try to formulate it unbiased.  Gerhardvalentin (talk) 09:26, 4 July 2011 (UTC)

Before we work on any bias, does everyone agree that the above describes the two position (possibly in a biased manner) If not, please try to provide a better description. Also needed, something short that we can use as a shorthand in the sentence "I am a _____ : I take the _____ position." This doesn't have to be completely descriptive, but maybe something better that Big-Endian and Small-Endian, (from Jonathan Swift) which is already used to describe different computer architectures. Guy Macon (talk) 10:03, 4 July 2011 (UTC)

okay, above I've added the two positions "door watcher" and "universalist" – other variants?  Gerhardvalentin (talk) 13:00, 4 July 2011 (UTC)

Guy Macon, Middle-Endian way of organising the data is also used in CS. Swift very exactly described what is going on here. The Big-Endianism goes back to prophet Lustrog, and Little-Endianism was introduced accidentally. "Positions" have been clearly stated in Gill's Stat Ned paper. Simplists versus conditionalists. The Middle-Endian way of thinking is showing the equivalence by good arguments -- my invitation.

>:A) Observers of door tickets: door watchers – The article is not about finding the answer switch >or stay, but about the correct technology in using Bayes and about conditioning on door >numbers, although completely irrelevant to correctly answer the famous question (being a "door >watcher").

For me the MHP has 2 aspects. 1st -- surprise (conditional!) that it is not 50:50. But once I learn that, I ask about OPTIMALITY of a strategy, and a particular action in particular decision making situation as a part of this strategy. The mathematical analysis can be performed under further assumptions. And the very criterion of optimality depends on the problem statement.

In the "famous situation" the conditional PROBABILITY is my least concern, in fact. Moreover, once you know that 1ss is overall optimal, this *implies* optimality in the famous situation, and conversely. Gill explained you that.

In my view the conditional probability is largely irrelevant to the MHP. Of course, you can write down 1/(1+q) and explain to laymen something about proportions and series of trials. Why nobody took care to do this btw? But it describes very special type of Monte's behavior which gives a little idea of what is going on..

Look: of million rounds of the game d1,d2,d3 appearBy N1,N2,N3 times. You pick always d1. Monte has freedom C times. If you switch from d1 you win N2 times, lose C times of C+N2 times in the famous situation. By all our good mind, switching from famous situation is better if N2 larger or equal C, agree? In the worst case C=N1. I have no idea what is C, if it comes from biased/unbiased coin or in some dependent on trials way -- it does not matter because C does no t exceed N1. But N1 and N2 are EXCHANGEABLE because door tickets are just to label doors. Therefore N2 in any reasonable sense not smaller than C. I win more games with switching from famous situation than with holding. This is what grandma or any gambler needs to know. And this is completely elementary and true subject to only symmetry among the doors. Instead of showing such insights, the article shows boring notation around Bayes formula, which I love as much as we all do. For me there is only one problem, to understand reasonable mechanisms and create a big picture. And your "positions" is the snow of winter twenty years ago. B) Yes switch now: universalists grant 2/3 – The article should clearly lay out to grandma and grandson why switching can never be beaten by any other decision, as always switching guarantees 2/3 >and no other method can ever do better (being a '"universalist").can you read Gill's passages carefully? Go to his homepage to see how it is done by counting three fingers. [User:Machtindex|Machtindex]] (talk) 14:00, 4 July 2011 (UTC)

The test of a first rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.

F. Scott Fitzgerald

The referent in the question of "what to name each position" could use some clarification. From my perspective, the different approaches to interpreting and solving the problem, such as combinatorial ones, Bayesian ones, game-theoretic ones, etc., are not really "positions" except in the minds of those who take a position on whether they are "right" or "wrong". That is to say, one can take a position about these interpretations and methods but they are not themselves positions. A perfectly reasonable person can take the position that they are all valid, even when they reflect mutually incompatible interpretations of what the question is asking.

Of more relevance here are the positions taken on approaches to covering the subject in an encyclopedia. I am tempted to name positions that have been taken during the history of this article as pedagogical ones, pedantic ones, and polemical ones; but such labels would probably be of little help in reconciling the positions. I just wanted to remark that treating the various ways of interpreting and solving the problem as "positions" strikes me as a polemical approach to covering the subject. ~ Ningauble (talk) 17:53, 4 July 2011 (UTC)

The various ways of interpreting the famous question, of "solving" the famous paradox and of finding the correct decision asked for can be found in ancient and newer sources, and editors here show mainly two (or more?) different views how to design and to present the article to the reader, and especially what attention to be paid to differing sources, and about their due WP:weight for the article and esp. for the reader. The editors here present mainly two different viewpoints (attitude, "position") in this respect. Is it undue to ask whether those two differing attitudes (or positions) in short can be characterized and named/labeled?  –  I don't see that as polemical.  Gerhardvalentin (talk) 19:42, 4 July 2011 (UTC)

I apologize if my post was unclear. I meant to draw a distinction between approaches to MHP, the problem, and approaches to MHP, the article.

The proximate reasons for my post were because (1) the title and introduction of this thread are not specific about the question on which positions are sought, and (2) much of the discussion in this thread struck me as revolving around the merits of two approaches to the problem. I can see that the intent was to address how to cover them, but my impression is that the discussion is still mired in assessing their merits and giving priority to one or the other. The broader background for my perspective, having watched this article off and on for 3 and a half years, is that the one constant in all that time is that tensions between approaches to the problem have impaired pedagogy, introduced pedantry, and inspired polemics.

My own position is neither "simplist" nor "conditionalist", which appear to me to characterize approaches to the problem. My position is that both approaches should be presented in a manner that is pedagogically clear and coherent. ~ Ningauble (talk) 21:42, 4 July 2011 (UTC)

Thank you Ningauble, your view is the only correct one, and I guess most of us welcome this position. Of course you are right. The paradox indeed offers a full range of different and really interesting aspects. And the "simple approach" should be shown, readily available to everyone if it is presented in a short and clear manner. I like to call it the "obvious way" to assess and to evaluate the underlying proportions.

And on the other hand of course the "conditional" view, preferably in odds form, should also be shown as the "academic solution of the paradox" for those interested in, although – as to prominent sources – this is not an indispensable requirement to solve this famous paradox. Neither "simplist" nor "conditionalist". But both aspects have to be shown, along with a full rage of other excellent approaches. Any stubborn tendency to unilateralism must be rejected, please forgive the harsh words. Literature is available, the article could profit so much. Once more: "Conditional probability" is just another way to solve the paradox. To show the fixed range of possible advantages (no disadvantages!) of switching here and now. Because the MHP does not ask for an exact "probability" for any special case, that never can be given for sure. It's sufficient to know about that "fixed range of advantages". Firstly and most of all the MHP asks for a decision, and I think to read you this way also, the correct decision can be based on a whole range of different approaches. Regards,  Gerhardvalentin (talk) 00:00, 5 July 2011 (UTC)

Ningauble, I appreciate your good intonations. The value of the struggles here is empty eggshell. Just do it folks: collect in one place, first perhaps, on experimental territory, all understandable good arguments. All ingenuity and insights. My only concern is the pedagogical issues - a spectrum of inspiration for the newcomers. Gerhardvalentin, Because the MHP does not ask for an exact "probability" for any special case, that never can be given for sure. Asks and do not asks depending on you. Why such dogmatism? Which ocean is bigger Pacific or Indian? You can answer this, then try quantifying. First you answer the questionoperationally, for the exact purpose you need, then you keep analysis, then you proceed to exciting extension and so on. To give you example of this line of thought let me start with the analysis of your FAMOUS SITUATION (so presuming door 1 chosen 3 revealed) having some unknown to me probability F. The prize is behind door 1 or 2 (events D1 and D2), so the disjoint union of events D1 and D2 covers F. The event D2 is subevent of F, therefore P(D2 , F)=P(D2)=P(D1) is larger or equal P(D1 , F). Read: In the FAMOUS SITUATION it is more likely to find the prize behind D2 than behind D1. (I wrote P(A,B) meaning intersection). This is all I need to favour switch. Please Note: I did not quantify ANY of the probs, nor even P(D1)=1/3. All what has been used is logic and the *symmetry assumption* P(D1)=P(D2) which you take for granted when say that the prize is located by fair 3-sided dice. The brilliant idea of symmetry for which you did not give to Gill any serious space. What I show you is MATHEMATICS on the level of everybody familiar with the concept of event mand addition rule. It gives you precise elementary answer to the famous concern in one line, in larger generality and without any quantification, without any "decision tree", concept of conditioning (is hidden, of course), without any extra letters like q and so on. I do not claim any originality on this elementary space, but I wish to tell you that when I hear that such passage cannot appear in the article because it was not published in the Annals of Probability --- I smile, because no serious mathematician will ever come to the idea that such triviality is publishable. Perhaps somebody 20 years ago appealed to cond prob, perhaps everybody refers on many "reliable sources", this site is *pedagogocal* and if highly educated editors find something interesting and simple -- it should be in the article, in my viewMachtindex (talk) 02:14, 5 July 2011 (UTC)

Machtindex, please read this: http://en.wikipedia.org/wiki/Help:Using_talk_pages#Indentation and in particular, please use the preview button to make sure your comments are properly indented before you hit save. Basically, look for ways your comments are formatted differently from everyone else's comments and make the so they are formatted the same. Also, my name is "Guy Macon. not "Guymacon", and the construct "@Username" is not usual on Wikipedia talk pages. Thanks!

Dear Guy Macon, I apologise for misspelling. Thank you for instructing me about indenting, I must confess this word always puzzled me as \noindent LaTeX command. The construct @username was copied from someones posting, which made me believe it is normal.

No problem. It is confusing. I have to thread/indent one way on Wikipedia, another way on USENET, another on Reddit, another on Slashdot, another on email written to people who have a clue, another on email from people who follow the Microsoft (spit!) "standard" of putting replies on the top, etc., etc. Then there are the habits I pick up from writing programs in assembler, embedded C, and python. The best way is to simply study what everyone else is doing and to do that. Guy Macon (talk) 12:23, 5 July 2011 (UTC)

Citation style

This article has an awkward citation style, could someone alter it? Template:sfn would be the most favorable. TGilmour (talk) 19:30, 1 July 2011 (UTC)

Editors are supposed to follow the existing style. Changing to a different citation style requires a consensus. We don't want edit wars over minor typography.

There is not a universally good choice for citations -- especially when some articles are referenced many times with changed page numbers. There are also gotchas when the same author published two works in the same year: 1999a, 1999b.

The current article uses a style known as Harvard referencing. Harvard referencing has the advantage of easy identification of notable sources; the reader doesn't have to look in the footnote to identify the author (e.g., Morgan). The downside is the Harvard reference breaks the text up more than a shortened footnote. SFN gets ugly when footnotes are shared.

The current article does not use templates for refs, but it does hyperlink refs with its own arcane system that uses a lot of HTML.

I've cheated a little bit when I've edited the article by keeping the Harvard format but using the {{Harv|author|year|p=123}} macro and the {{Citation ...}} macro. Generally, the appearance is approximately the same, and there's a hyperlink. But it also means I have to fix all the existing links in the article. A citation bot will also periodically edit a citation macro (sometimes with poor results, but they've gotten a lot better).

Some editors are adamantly against the citation templates because they do not follow common editorial standards (such as order of fields; placement of initials for subsequent authors). I don't care about that; I'd rather see consistency across articles. (The Citation format for a patent currently drives me up the wall.)

Changing article citations is a lot of work for little gain. A large change can break a lot of links. Generally, WP discourages changes that don't impact the printed appearance of an article.

Consequently, I would continue to use Harvard citations because that was the initial choice.

I would support using Harv macros and changing the current references to Citation macros slowly over time because it does not significantly change the citation system. I would only change the citations when I was adding a reference.

Glrx (talk) 20:22, 1 July 2011 (UTC)

Yet another advantage of SFN (or Harvnb, it doesn't matter that much) is that it indicates the page unlike this style of citation. TGilmour (talk) 21:17, 1 July 2011 (UTC)

• Comment. User:TGilmour has been blocked indefinitely as a suspected sockpuppet. Glrx (talk) 19:20, 7 July 2011 (UTC)

SFN vs Harvard referencing

Thoughts are divided so I decided to open a poll.

Other

*Oppose major change per WP:CITEHOW and my comments above. Glrx (talk) 03:31, 3 July 2011 (UTC)

There is stated that it isn't allowed unless a large consesus is gained. TGilmour (talk) 16:43, 3 July 2011 (UTC)

TGilmour: you should not edit the content posted by others on a talk page. I did not strike through my post above. I voted my opposition for a consensus that involves radical change. I do not want a supposed consensus for Harvard referencing to be interpreted as permission to make major edits to an article that results in little content change. Glrx (talk) 18:09, 4 July 2011 (UTC)

Harvard referencing

• Support (but it is only a weak preference); it makes for easier reader, I think. Richard Gill (talk) 07:19, 2 July 2011 (UTC)

Shortened footnotes (SFN) or Harvnb

• Support I think, this style is a better one. TGilmour (talk) 21:16, 1 July 2011 (UTC)

Can we agree on what to name each position AND NOTHING ELSE?

OK, I am starting a new section on what to name each position, because the last section filled up with material that, while interesting, having nothing to do with that specific question. This time I am asking, if you don't have a specific proposal to a name or a short comment on another such proposal, please pick another section.

Here are copies of the parts that seem to be answers to the question (shortened and paraphrased, feel free to edit if I got your position wrong): My comments follow each paraphrase

From Gerhardvalentin: "door watcher" and "universalist." Door watchers are observers of door tickets – The article is not about finding the answer switch or stay, but about the correct technology in using Bayes and about conditioning on door numbers, although completely irrelevant to correctly answer the famous question (being a "door watcher"). Universalists saye article should clearly lay out to grandma and grandson why switching can never be beaten by any other decision, as always switching guarantees 2/3 and no other method can ever do better (being a "universalist").

Is either of these the same as "incrementalist"? Guy Macon (talk) 05:03, 5 July 2011 (UTC)

From Machtindex: Big-Endian, Middle-Endian, Little-Endian.

I could not parse a short description for each term, nor is it clear that the terms are intended to be answers to my question. This is not an invitation to post another 1000 word essay, but three short (less than 10 words each, 20 at the max) descriptions would be helpful. Guy Macon (talk) 05:03, 5 July 2011 (UTC)

From Ningauble: "I meant to draw a distinction between approaches to MHP, the problem, and approaches to MHP, the article. ... the title and introduction of this thread are not specific about the question on which positions are sought."

I meant approaches to the article. In particular, I meant "that which we have been arguing about for many years without reaching a consensus agreement." To be more specific, I am looking for something I can use when entering in to Wikipedia Content Dispute Resolution instead of saying "Position on how this article should be written held by A" and "Position on how this article should be written held by "B". Guy Macon (talk) 05:03, 5 July 2011 (UTC)

I see three positions in the literature and three corresponding positions concerning how the article should be organised. I would call the positions: simplist, conditionalist, strategist. Here is a little essay on the topic. Richard Gill (talk) 08:29, 5 July 2011 (UTC)

essay on three positions: simplist, conditionalist, strategist

"Simplist", "conditionalist", "strategist". Within the enormous literature on Monty Hall problem, one can find two extreme points of view concerning how the problem should be solved. I'll call them the simplist's solution and the conditionalist's solution. The opinions of editors of the wikipedia article on MHP are similarly divided as to how to deal with this perceived conflict. Note: not all sources see a conflict, and not all editors see a conflict either. The strategists see ways to reconcile the points of view of simplists and conditionalists. They think that both simplists and conditionalists see things too simply. I'll come back to them later. Simplist editors think that the article should concentrate first on the simple solutions and not even mention the conditional solutions, or the controversy, till later. Conditionalist editors think that the controversy should be mentioned as soon as possible to prevent readers from assuming that there has been no criticism of the simple solutions. They want to see conditional solutions presented alongside of simple solutions, at the outset. Strategist editors do not think there is a controversy. The simple solutions assume less and derive a sensible reason for switching. The conditional solutions assume a lot more and derive a better still reason for switching. There are nice bridges between the two which make both easier to understand. Simplist solution: MHP is a simple brain-teaser. The intuitive answer "it doesn't make a difference, you might as well stay with your initial choice" is wrong. You have to realize that 2/3 of the times you initially pick a goat and then by switching you get the car. Only 1/3 of the time do you initially pick a car. So switching doubles your chance of winning the car from 1/3 to 2/3. That's it. Conditionalist solution: MHP is a conditional probability problem. You are asked if you want to switch, after you chose a specific door (Door 1) and the host opened a specific door (Door 3). The question is, how often would you win the car by switching in this specific situation. It's given that the player chooses Door 1. There are three possibilities for the location of the car. 1/3 of the time the car is behind Door 1. The host could now open Door 2 or Door 3. Half of those times Door 3 is opened. So 1/3 x 1/2 = 1/6 of the times, Door 3 is opened and the car is behind Door 1 (A). 1/3 of the time the car is behind Door 2. Door 3 is then always opened (B). 1/3 of the time the car is behind Door 3 but then Door 3 is never opened. Altogether, Door 3 is opened in situations (A) and (B) which is 1/6+1/3 = 1/2 of the time. The car is won by switching in situation (B), chance 1/3, but not in (A), chance 1/6. So given that the host opens Door 3 (which happens half of the time), the player wins by switching twice as often as he loses by switching (1/3 versus 1/6). This makes the chance of winning by switching given the player chose Door 1 and the host opened Door 3 equal to 2/3. That's it. Yet other sources have other solutions. Some sources are "catholic", presenting a variety of solutions without commenting on the difference. Others try in various ways to build bridges between the simple and conditional solutions. This goes back to the initial eruption of the controversy between simplists and conditionalists, Vos Savant versus Morgan, Chaganty and Rao: see the published discussion contributions on Morgan et al. In recent publications --- inspired by the wikipedia conflicts --- Gill and Gnedin emphasize that the the point of computing probabilities in MHP is in order to make a decision. From their point of view, the simple solution shows that "always switching" is good. The conditionalist solution shows that "always switching" is best. Gnedin has come up with an alternative proof that "always switching" is best, which does not make use of conditional probability. Strategist solution: From the simple solution we know that "always switching" wins the car 2/3 of the time. Such a player loses the car 1/3 of the time. Consider a player who would however prefer to stay in at least one situation. For concreteness, suppose the player chose Door 1 and plans to stay with his initial choice if the host opens Door 3. Now given that the player chose Door 1, if the car were actually behind Door 2 the host would be forced to open Door 3. Our player, who won't switch in this special situation, will lose the car whenever it is behind Door 2: that happens 1/3 of the time. So this player is also going to lose the car (at least) 1/3 of the time. We see that however you play, you're bound to lose the car at least 1/3 of the time. "Always switching" gives you the car with biggest possible chance, 2/3. That's it. It's interesting to note that the three solutions presented here make different assumptions. The simplist solution only assumes that the initial choice has 1/3 chance to hit the car. This could just as well be true because the player chose his door at random, as that the game-organisers hid the car at random. The conditionalist solution assumes that the car is equally likely to be behind any of the three doors, and that the host is equally likely to open either door, if he has a choice. The strategist solution assumes that the car is equally likely to be behind any of the three doors, but not necessarily that the host is equally likely to open either door when he has a choice. As a bonus we can also conclude in this situation that the conditional probability of winning by switching, given door chosen and door opened, can never be against switching. From the mathematical point of view this means that the three solutions presented here are strictly speaking different. It is not the case that one is better than the other. The more you assume, the more you can conclude. None should be rejected. Understanding all three is a much bigger understanding of MHP than understanding just one of the three. My personal opinion is that twenty years on from the fight between Vos Savant and Morgan--Chaganty--Rao, this conflict belongs to history. Editors of the wikipedia page on Monty Hall problem are hopefully enough "on top" of their material that they can present a variety of solutions alongside one another, showing advantages and disadvantages of all, and leaving it to intelligent readers to choose what appeals to them best, if a choice must be made at all. Richard Gill (talk) 08:29, 5 July 2011 (UTC)

essay on (one? two?) position(s): symmetrist, combinatorialist

"Symmetrist", "combinatorialist". Richard, the list is incomplete. There is one more approach you missed to mention. COMBINATORIALIST, which is basically the same as SYMMETRIST. This position is dominant mathematically, as it implies whatever you wish to derive without probability assumptions. Without conditional probability, without probability at all and if with probability then with many dependencies. I do not touch pedagogical issues at the moment. I am mathematician looking for proofs from The Book. Some have been shown above, and you will see more, just breath in and out. Machtindex (talk) 11:00, 5 July 2011 (UTC)

OK, I have ten proposed names so far (in alphabetical order):

Big-Endian
Combinatorialist
Conditionalist
Door Watcher
Little-Endian
Middle-Endian
Simplist
Strategist
Symmetrist
Universalist

Now I know that there are not ten separate ways of writing the MHP Wikipedia page, each with proponents who reject the other nine. I know there are two, and there may be three. So let's work together to discard any duplicates, subtle distinctions, and positions that have zero proponents (including past participants who are currently banned or have given up).

Remember, this is just for the narrow purpose of taking the fundamental XXXXXXX vs. XXXXXXX content dispute that we have been debating for the last few years to Wikipedia Content Dispute Resolution. It has no other purpose and does not require academic rigor. It does require that everyone who has been debating the question of how to write the MHP article agree that "Yes, I will agree to call my position on how to write the article XXXXXXX." This is necessary but not sufficient to resolve the content dispute. Guy Macon (talk) 12:14, 5 July 2011 (UTC)

Guy Macon, I like what you say. It seems we are approaching consensus. However, I am now a bit concerned that whatever we create can be re-edited by future even more cute editors. What do you think?Machtindex (talk) 12:38, 5 July 2011 (UTC)

That's the beauty of completing the Wikipedia Content Dispute Resolution process. Once complete, we will have an authoritative answer that says "The consensus of the larger Wikipedia Community is that the MHP page will be edited according to the principles of XXXXXXX and not YYYYYYY." At that point, anyone insisting on editing it according to YYYYYYY will have his edits reverted. He can, of course, go through Content Dispute Resolution again and seek a new consensus, but until he does, the argument will have a clear winner and a clear loser. I have no idea which side will win; both have reasonable arguments. This will settle the issue for good, and put an end to the endless debates. (PS: use your "show preview" button to check for indentation errors when you reply) Guy Macon (talk) 15:31, 5 July 2011 (UTC)

If the purpose of this section is to identify positions "on how this article should be written" for the purpose of entering in to Wikipedia Content Dispute Resolution then the two most salient positions for that purpose are:

• Exclusivists – those who take the position that some "correct" approach to MHP (the puzzle) is superior, and that others should be omitted or presented in a way that denigrates them as inadequate for solving MHP or deprecates them as unnecessary for solving MHP.
• Neutralists – those who take the position that all noteworthy approaches to MHP (the puzzle) should be covered from a neutral point of view.

I reiterate my point from the previous thread that some of the so-called "positions" identified above are approaches to MHP (the puzzle). To label someone as an adherent to one of those positions, as a position on how this article should be written, is to label that person an Exclusivist.

For Neutralists the content question is not which approach to MHP (the puzzle) is best, it is how to cover each of the noteworthy approaches from a neutral point of view and to organize them into a balanced and coherent article. Neutralists do face challenges in achieving this, and in deciding whether some approaches are too obscure to merit inclusion, which is not the same as adopting one approach as best. I have some opinions about how best to cover some of the approaches, but I am not even going to mention them if it means being put in an Exclusivist box associated with any of those approaches. ~ Ningauble (talk) 19:31, 5 July 2011 (UTC)

Thanks! I am going to watch for a while to see what agreement or criticisms others here have before commenting. Guy Macon (talk) 20:17, 5 July 2011 (UTC)

@Ningauble, I'm afraid every editor will solemnly declare that they are a Neutralist. Especially since the editor who insisted that the simple solution was downright wrong (and as a corollary, all sources giving it, are not reliable) got banned for a year.

But anyway, I'ld like to draw your attention to two encyclopedia articles on MHP which I think do take a Neutralist position and which present simple and conditional solutions side-by-side: MHP on StatProb and MHP on citizendium.org. You will recognise my hand in both of the them. The citizendium article grew out of wikipedia's, the StatProb article grew out of citizendium's. I'll probably rewrite both in the near future now that we have even simpler solutions thanks to A.V. Gnedin's recent publications. Richard Gill (talk) 11:35, 6 July 2011 (UTC)

I looked at those a few months ago, and may give them another glance. ~ Ningauble (talk) 17:12, 6 July 2011 (UTC)

It seems to me the most obvious "position" or "MHP faction" has been overlooked completely: the eternal discussionist :-)--Kmhkmh (talk) 12:05, 6 July 2011 (UTC)

+1. ~ Ningauble (talk) 17:11, 6 July 2011 (UTC)

A better way forward would be to abandon creating a taxonomy of Exclusivist positions, which are not going to fare any better in Dispute Resolution than they did in Arbitration, and focus instead on identifying specific differences between Neutralist alternatives, with specific attention to objectives for overall framing of the article and to ways of presenting the relationships between approaches to MHP. ~ Ningauble (talk) 17:11, 6 July 2011 (UTC)

The various positions on the content of the MHP article were ignored by arbcom, and it was repeatedly stated during the case that it is not the role of the arbitration committee to rule on content disputes. Arbcom in about conduct, not content. Content Dispute Resolution is designed to resolve content disputes, and you have no reason to believe that is will fail without trying it. Guy Macon (talk) 20:25, 6 July 2011 (UTC)

Although Arbcom's actions are indeed about conduct, not content, in this case their principles and findings would appear to deprecate extreme Exclusivist perspectives. I am not saying DR will fail, but that positions that are at variance with principles 3 and 4, e.g., cannot be expected to fare well. My real point was to say that asking who likes which approaches to the problem, and characterizing them as opposing positions, is probably not the best path toward finding a way to present the different approaches together in the article. I like most approaches to common interpretations of the problem. ~ Ningauble (talk) 15:52, 7 July 2011 (UTC)

Alas, we have had multiple attempts at making this article into something that everyone accepts, all of which have failed to achieve that goal. This does not mean that the next attempt will fail as well, of course -- maybe the latest suggestion by Richard Gill will end up being acceptable by all -- but I am no longer willing to put my plan on hold again and again while Yet Another Attempt At Consensus plays out. Instead, I am moving forward with a willingness to drop the Content Dispute Resolution case if, at long last, consensus is reached. All evidence so far suggests that there really are opposing positions and there really is an intractable content dispute, and I am moving forward on that basis (with the next step being naming the positions). Nothing would make me happier that to be proved wrong by content that is acceptable to all, but I am not holding my breath.

Any opinions on content gleaned from arbcom can be safely ignored; it it not the role of arbcom to rule on content disputes, just user behavior. As I have said many times, nobody here is obviously right or obviously wrong, and Content Dispute Resolution might end up favoring any of the positions. It will mostly depend on how well each proponent crafts a logical policy-based argument supporting his preferred content. Guy Macon (talk) 18:35, 7 July 2011 (UTC)

Again, I am not saying that DR is doomed to fail. See at "My real point was..." above. I am saying that defining indefensibly narrow positions is an invitation to waste time on non-starters. Go ahead and include them at DR if you can find supporters for such extreme positions as using only the simple approach or excluding the simple approach. Please forget I mentioned that Arbcom happened to point out some fundamental policies. Their advice is not needed. ~ Ningauble (talk) 19:21, 7 July 2011 (UTC)

another digression

Dear all, I am close to stating my position. First of all, I am ready to retreat for a while from the term Big/Little Endian controversy, as this is basically covered by the simplist-conditionalist classification by Gill. So probably I share views with the following ones Combinatorialist, Conditionalist, Simplist, Strategist, Symmetrist, Universalist, Educator, Probabilist, Statistician (classical and Bayesian), Novelist, Behaviorist, Geometer, Optimisator, Information theorist, Machine learner Operations Researcher, Dynamical Programmist, Complexity theorist, Analyst Algebraist (conditionally at the moment) I have good (in my view) arguments for each of this views. My further suggestion is to start working. This is a long way to go, as I see it at the moment. I hope to be given opportunity to indeed represent each of the mentioned positions in the final product. Richard has done some preliminary work, which seems to be a good start. I lost a bit overview over participant of the adventure. May we compose a preliminary list. I understand we have Gill, Guy Macon, Rick Block, Machtindex, Gerhardvalentin, Ningauble, Kmhkmh Please correct/complement my list if something is wrong. Machtindex ([[User ::talk:Machtindex|talk]]) 19:58, 6 July 2011 (UTC) Please take a close look at the line starting with "Dear all, I am close to stating..." Now look at the line starting with "for a while from the term..." Do you see anything -- anything at all -- that seems strange or different between how those two lines are indented? Do you see anyone else writing comments that look like that? Now formulate a theory as to why the indentation is the way it is, and test your theory. Guy Macon (talk) 20:18, 6 July 2011 (UTC) Dear Guy Macon, thanks for the bright suggestion. I am rather bad with all this things. Instead of creating a theory I try first some experimentation. If it is fine, theory of identiation is my least concern for a while. What would you think, suppose Monte knows 3000 games in advance, can he help you, by opening doors, to guess the prize almost all the time? 3/4 is easy, perhaps 7/8, but 1 minus epsilon...?Machtindex (talk) 21:20, 6 July 2011 (UTC) Thanks for doing a better job on indentation. It is important because it makes your comments easy to follow. I am not going to answer the above question, because it appears to have nothing to do with the specific question of what to name each position on how to write the article. I really need comments on that specific question. Guy Macon (talk) 06:13, 7 July 2011 (UTC)

As I recall, the content dispute concerned whether or not the initial sections of the article, which clearly must concentrate on the most widely accessible and broadly interesting material, should also include either (a) warnings that some authorities consider the popular (simple) solutions definitely wrong or (b) should actually include some of the more sophisticated solutions. In the past, the article used approach (a). This led to constant fights and complaints. If we choose (a) again, we will have constant fights and complaints in the future. So why not try (b)? Include among the simple solutions also a simple (non technical) version of the conditional solution. Use the simple solution and symmetry as a gentle way to get a stronger conclusion. Explain in a positive sense what it brings. Don't express it as a criticism. Use it as a positive inducement to the reader, to learn about conditional probability. Tell them they can learn more about this topic later in the article, if so inclined.

Of course (c) writing the initial part of the article as if advanced solutions don't exist, would also lead to eternal conflict. It seems to me that the only thing that hasn't been tried yet (but I tried it on citizendium and on StatProb!) is the genuine compromise (b). So far we never tried that because most people were firmly entrenched in (a) or (c).

So to answer patient Guy's question, if that is still the dispute, why not call the three positions (a) EU Directive Health Warnings, (b) The Wikipedia Way, (c) Apartheid. And let's choose (b).

Then later in the article there can be a small section about Controversy, and one of the controversies can be the Vos Savant vs Morgan story. Along with criticism of the criticism. (Blatant advertisement of OR). Richard Gill (talk) 12:04, 7 July 2011 (UTC)

Thanks Richard. The issue of health warnings for putative Lies-To-Children clearly identifies one of the main issues on which people have taken opposing positions about how to cover "simple" approaches. My own opinion is that simple approaches reflect a notably reasonable interpretation of what it means to say "say, door #3" rather than just saying "door #3," and should not need warnings, qualifiers, or justifications; that for pedagogical clarity the complications belong in the advanced treatments, not the simple ones; and that giving due weight to different approaches should not result in giving undue prominence to minor controversies. ~ Ningauble (talk) 16:42, 7 July 2011 (UTC)

Proposal: name the two content positions "with health warnings" and "without health warnings." Anyone oppose? Hopefully with a counter-proposal? Guy Macon (talk) 18:39, 7 July 2011 (UTC)

"Health warning" is pejorative and these two alternatives apparently exclude Richard's (b) approach, above (which is consistent with what I have been arguing for, for a long time). As I see it, the two positions we're actually discussing are "conditional probability based solutions presented as equally valid alternative to simple solutions" (i.e. Richard's (b), above) and "conditional solutions subservient to simple solutions" (i.e. Richard's (c), above). I don't see anyone here arguing for Richard's (a), above (although I think there are folks who think this is what I'm arguing for - I'm not). -- Rick Block (talk) 22:59, 9 July 2011 (UTC)

Ok, "health warnings" could be too divisively hyperbolic for what may only be a difference over nuance. (The discussion may have finally moved beyond the point when some contributors argued for explicitly deprecating "simple" treatments.) Still, I think there is an ongoing difference of opinion that arises from mixing and matching methods of solution with interpretations of the question. The idea for a unified solution section is a very appealing approach to simplifying the article but, in burying the difference between different interpretations of the problem, I am afraid it presents real challenges. E.g., the draft language at "The middle way: synthesis and synergy, not opposition" above could be taken to imply that formal proof must entail conditional probability and that simple solutions are merely heuristic approximations. (I do not mean to pick of the drafter of that proposal, it is a very good attempt.) I will post further thoughts on interpretations of the problem in another thread below. ~ Ningauble (talk) 17:58, 10 July 2011 (UTC)

Wikipedia:Ignore all dramas

Difficulty In Reaching A Unanimous Consensus If I understand the findings and remedies from the arbcom, there remains one participating editor that had been exercising ownership violations for quite some time. His primary disagreement with the consensus, NPOV violation, is unchanged. Is it appropriatee to declare a stalemate requiring dispute resolution in this instance? Or should the existing arbcom finding and remedy be given greater consideration? 166.216.194.165 (talk) 16:49, 8 July 2011 (UTC) There has been zero ownership or any other misbehavior since the arbcom. I have been looking carefully for any ownership, incivility, assuming bad faith, etc. so that I can warn that user that he is slipping into a bad habit. Zero problems. Not even a hint of misbehavior. The arbcom remedy appears to be 100% successful. If you are under the impression that arbcom rules on content disputes you are mistaken. Arbcom only deals with user behavior. There is no consensus on the content dispute. If there was, I would be telling whoever does not accept the consensus to do so, and would treat any refusal to do so as a behavioral problem to be dealt with in the usual way. BTW, "Unanimous Consensus" is not what Wikipedia requires. Go to Wikipedia:Consensus and read the section on "What consensus is." What we cannot do is ignore a legitimate claim of a NPOV violation. We need to determine whether the violation is real. There are good arguments either way. Do you have an alternative to declaring a stalemate requiring dispute resolution in this instance? Guy Macon (talk) 17:48, 8 July 2011 (UTC) This page is a mess. Rick Block has consistently argued against the long term consensus that the simple solutions to the simple brainteaser that most people get wrong should be the main thrust of the article. I proposed a compromise which had support from a clear majority of editors, which was that we should follow the lead of most good text books and present the simple solutions first (with no health warnings) followed by an equal weight being given to the, so called, conditional solutions. This has been ignored despite getting majority support. NPOV is irrelevant because I am suggesting that we present all forms of the problem and solutions fully in the article, starting with the simplest and by far the most notable. Ricks claim of 'structural POV' is simply absurd. Let us put this article back to the way that the majority of editors want it and the way that the vast majority will understand it. Start with a clear and simple explanation of why the answer is 2/3 and then address the various mathematical subtleties. Is there anyone other that Rick who disagrees with this? Martin Hogbin (talk) 22:58, 8 July 2011 (UTC) Hallo Martin, I am pleased to see you here. I think I agree with you basically. The MHP paradox is ::absurd by itself, related to the first surprise caused by the fallacy of "symmetry", and then by ::incapability to view the situation under slightly different angle. The discussion going on here is absurd squared, scholastics without limits. I came up with proposal simply compose a reasonable document which could be then edited and discussed. Instead, I see discussions which I fail to follow. My question therefore, who is interested to work on the page personally, not engaging in the further nonsense. My current understanding, however, that MHP makes sense not only without cond. prob., but without probs at all, just combinatorially two out of three that's it, and same for conditional probs. The "paradoxes" simply disappear. Who is interested to work on the page seriously, not engaging in any further debates? If there is no answer, then perhaps there is a little to seek here, or let us split and look for a good site elsewhere.Machtindex (talk) 23:38, 8 July 2011 (UTC) Martin, I think it would be better if you would say "simply explain why the answer is switch". Then you could expect unanimous agreement. It's by saying "the answer is 2/3" that problems are created. What, exactly, is 2/3? Don't raise that question. The answer is "switch". One of the reasons for switching is that you'll get the car every time you initially pick a goat, and everyone will agree that that happens 2/3 of the time. Various authors give various other good reasons for switching. The better the reason, the more assumptions have to be added. Machtindex: your new ideas are beautiful but hardly yet known in published literature. For that reason you should consider promoting them on citizendium.org or on StatProb.com. Those are alternative online encyclopedias. Anyone can contribute to them, but both have peer review, and editors are not anonymous. StatProb is hosted by Springer but run by all the major statistics and probability societies. The advantage of doing it that way is that once your contributions have been accepted there, they become "reliable sources" for Wikipedia editors. Richard Gill (talk) 11:08, 9 July 2011 (UTC) I do not think 'switch' is good enough, neither does it represent the best known solutions to the problem. The point that people find hard to accept is that not that there might some slight advantage is switching but that you double your chances (using most people's intuitive understanding of the word) of winning if you switch. That is what the simple problem is all about and what we should start with. That is not to say that we should not explore all the interesting mathematical subtleties in full later. Martin Hogbin (talk) 14:04, 9 July 2011 (UTC) Yes, I am aware that a 'consensus' need not be unanimous. It seems, however, that that is the perhaps unstated requirement for this article. As suggested above, it might be enlightening to see if a consensus could be reached sans the editor who continues, despite the arbcom findings to be the outlier. The 'Wiki Way' suggestion from Mr. Gills is probably approach to seek consensus on. As for the Ownership violation I mentioned, unlike those obvious and immatetial civility issues you mentioned, ownership works best when it is presented in a friendly, authoritative and subtle fashion. It was years before arbcom ruled on the editor of interest here, and many admins and editors were unable to detect it, despite it being called to their attention. I read no admission nor remorse nor comittment to change by that editor in the arbcom. In fact, what may have been his only comment was to imply a miscarriage of justice. 166.216.194.38 (talk) 15:20, 9 July 2011 (UTC) Again you misstate the role of arbcom, despite it being repeatedly explained to you that arbcom does not rule on content disputes. Your statement parses thusly: "it might be enlightening to see if a consensus [on the content] could be reached sans the editor who continues, despite the arbcom findings [on user behavior] to be the outlier." It assumes something that is not true, that the arbcom rules on content. I previously stated that I saw no hint of misbehavior, and would have warned any editors if they were slipping into bad habits. Sadly, I must now warn you that you are starting to exhibit behavior that, if it continues, will be a behavior problem. You are on the verge of WP:IDIDNTHEARTHAT. Please stop implying that arbcom rules on content disputes. Concerning your other questionable claims, we have had multiple counts and in none of them was there a lone editor disagreeing with everyone else. Your theory of ownership so subtle that editors are unable to detect it does not match the description in WP:OWNERSHIP, arbcom has no requirement for remorse (this isn't a re-education camp run by the thought police) and it is perfectly acceptable to state that you think an arbcom ruling to be a miscarriage of justice. I am going to ask a question. You are not required to respond in any way. As an IP editor with a grand total of two edits and a history going back two days, you seem to be very familiar with the arbcom case. Is there anything you wish to voluntarily reveal about a previous identity on Wikipedia? Again, you don't have to answer. Guy Macon (talk) 16:55, 9 July 2011 (UTC) My interest is in reading a coherent article that is accessible to Wikipedia's readers. I have no interest in being lectured by you, threatened by you, or having my comments re-stated by you. I suggest you see if there is a consensus for Mr. Gills' Wiki Way' proposal. 166.216.194.56 (talk) 17:20, 9 July 2011 (UTC)

Probability or Decision?

I'ld like to retrieve one small item from the above hidden stuff. Martin's remark "the answer is 2/3" and mine, "no, the answer is switch". I did not mean to imply that the 2/3 should be made completely invisible. Of course one must emphasize that a good *reason* to switch is because the probability is 2/3 that you'll get the car this way. It is even a very very good reason to switch, since it is can be shown (if all doors are initially equally likely to hide the car) that you can't do better; in particular, complicated strategies of sometimes switching sometimes staying depending on specific door numbers lead to a smaller chance of success! (Morgan et al; Gill).

I hoped that this remark of mine would help show that by careful choice of words it's possible to write sentences which many more editors could agree with. And which everyone can agree are not misleading and which everyone can agree reflects what is in the literature.

Remember: Vos Savant and Whitaker ask what the player should do. Introducing probabilities into the story is one way to set about finding justification for an action. The phrase " the probability" leads straight to controversy, at least to disagreement (both among editors and in the literature). The phrase "what action should the player take" leads to concensus (both among editors and in the literature).

It's just meant to be a small, practical concensus building suggestion! Richard Gill (talk) 14:45, 10 July 2011 (UTC)

PS I do agree that there are a couple of sources who do say "Craig Whitaker is asking for such and such a probability". That's their opinion. But it's not in Craig's words nor in Marilyn's paraphrase of them. Richard Gill (talk) 14:51, 10 July 2011 (UTC)

K&W problem statement

The most notable, by far, problem statement is that by Whitaker/vos Savant and that is the one that should precede vos Savant's solution. Virtually no one has heard of the K&W problem statement. It is useful for giving the most common interpretation of the problem before more advanced discussions so it should come later down in the article. Martin Hogbin (talk) 22:41, 9 July 2011 (UTC)

The vos Savant statement is quoted in the lead. We can quote it again (a paragraph later), but Glrx (for one) objects to the duplication. No one has heard the K&W problem statement, but it presents the problem as both 1) nearly everyone understands it, and 2) nearly all sources interpret it. Presenting solutions without presenting a clear problem statement seems like a bad idea. What is that quote you're so fond of - "Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem" -- Rick Block (talk) 23:15, 9 July 2011 (UTC)

I don't object to the duplication of the simple problem statement early on. I believe the problem stmt in the article intro is recent enough that it need not be repeated, but the repetition is not be bad. Glrx (talk) 23:21, 9 July 2011 (UTC)

The K&W formulation is common in the pedagogical literature of statistics and probability. I disagree that "nearly everyone understands it" this way, and also that "nearly all sources interpret it" this way. By emphasizing the K&W formulation one creates a formulation biased to the conditional solution. However it is of course fine to list the assumptions which are commonly introduced as long as one emphasizes that not all of the them are used in every solution. For instance, Monty Hall's own solution (quoted with approval by Selvin) corresponds to the host-side picture that the location of the car is fixed and the door picked by the player is random. That is also the solution given in the text-book by Georgii, and in the text-book by Haggström. It's the solution from game theory and economics (Nalebuff). It's a solution which has repeatedly been proposed by new editors on wikipedia. And it's one of many solutions mentioned in the works of yours truly. Richard Gill (talk) 14:25, 10 July 2011 (UTC)

(e/c) There should be a simple problem statement in the beginning of the article; an extended problem statement should be deferred until the MHP variations are addressed much further down. Most people understand the relevant details from the W/vS description. A detailed problem statement up front lacks context and motive, so it is a distraction to the reader. Some solutions need to make the details explicit (e.g., q in a decision tree), but those details should be addressed when discussing those solutions. Glrx (talk) 23:17, 9 July 2011 (UTC)

I agree variations should be deferred to a variations section further down (as they are in the current article), but deferring (for example) that the host MUST make the offer to switch has nothing whatsoever to do with variations (well, there are game theoretic variations where these assumptions are relaxed) and rather than "distract" the reader, simply clarifies the problem. These clarifications were mentioned by Selvin in his second letter, and (most of them) by vos Savant in her subsequent columns. They are certainly not new in any sense with K&W and, per Barbeau (and others) are part of the "usual" problem. There are people who justify the 1/2:1/2 answer on the grounds that the MvS description does not preclude the interpretation that the host opens a door completely at random and (in this instance) does not reveal the car. IMO, omitting these clarifications is just being sloppy. -- Rick Block (talk) 01:00, 10 July 2011 (UTC)

Reply to Glrx: The point is that there are many different "extended" problem statements. That is where the richness of the literature comes from, and also many of the struggles between editors. Everyone has to realise that according to the literature there is not one preferred formulation, even if each individual editor may have a personal preference.

Reply to Rick: yes, agreed. It's essential to make clear right at the start that the host knows where the car is and is certainly going to open a door (a) different to that chosen by the player and (b) revealing a goat. (It's not part of the basic problem description *how* he does this - at random or not - since the point of solving the problem is coming up with reasonable assumptions in order to get a solution.) The situation where the host might or might not open a door, and/or might reveal a car as well as a goat, are variations to be kept separate and studied later - some amusing, some pedagogically important. See especially Steinbach's beautiful solution of the situation where we do not have this clarification and don't know whether Monty is trying to help you or to hinder you. Richard Gill (talk) 14:30, 10 July 2011 (UTC)

Rick, you are right that already ab initio it must be very clear on which issue the article starts, and that variants belong to a later variants section. And I add that imho the only correct "solution" of the MHP is the correct decision "yes, switch". That must be and has to remain the central theme and the core area of the article, and entertaining additional considerations should not disturb that main focus. Gerhardvalentin (talk) 15:45, 10 July 2011 (UTC)

"Nearly everyone understands it this way" is not my claim, it's what K&W say. Their problem statement is perhaps biased to a conditional solution, but regarding this they say (again, not I say, they say) nearly all people interpret the MvS problem to be about the specific (conditional) case where the player has picked door 1 and the host has opened door 3. If you want to argue either of these, please find some published source that disagrees with them (presumably from the experimental psychology literature). Lacking any such sources all you're stating is your own opinion, which is (of course) biased to your preferred approaches.

Hold it Rick. Yes, I know K&W say this. And of course everyone understands that you must, if only by way of example, answer what you would do if you had chosen Door 1 and the host opened Door 3. To characterize this understanding, you use the word "conditional"! As if it to imply "everyone understands you must solve the problem with conditional probability"! I don't think K&W say that, and moreover, it is manifestly untrue. Richard Gill (talk) 06:48, 11 July 2011 (UTC)

The point that the host is assumed to choose between two "goat doors" uniformly is what forces the problem to be symmetrical, and ensures that the probability in any specific case is the same as any other in which case the question "what is the probability of winning for a fixed strategy of switching" has the same answer (2/3) as the question "what is the probability of winning by switching if the player has picked door 1 and the host has opened door 3". The claim that nearly all sources interpret the problem per the K&W statement is also not mine, it is Barbeau's (and many others). Yes, there are sources that specifically interpret the question to be about the strategy of switching, but this question is a variant (compared to the bulk of the literature). The problem is the popular sources (starting with MvS) state what is apparently the conditional probability question, don't specifically say what question they're addressing, but then proceed to clearly address the strategy question rather than the question K&W say nearly all people interpret the problem to be asking. Again, this is not my claim - it's something Rosenhouse says in his book - "... continuing with their [Morgan et al.s'] lengthy essay makes clear that their primary issue with vos Savant is her shift from what they call the "conditional problem" as posed by her correspondent (in which it is stipulated that the contestant always chooses door one and the host always opens door three), to the "unconditional problem," in which we stipulate only that after the contestant chooses a door, the host opens one of the goat-concealing doors. She did, indeed, make this shift ..." (emphasis added). Read this again, carefully. Rosenhouse (not me) is saying vos Savant presented question A ("what is the probability of winning by switching if the player picks door 1 and the host opens door 3") and answered question B ("what is the probability of winning for a strategy of switching"). These have the same answer, but only with the usual assumption that the host chooses between two goat doors uniformly.

All these references are people from probability and statistics who follow the line set out by Morgan et al. who already twisted Marilyn's words for their own purposes. When you talk about "the bulk of the literature" you refer to the bulk of the literature in academic statistics and probability teaching. Which just copies and recopies what earlier people did. It's how unthinking people in this field traditionally approach this problem, which has become a fixture in chapter 1 of elementary probability texts. You leave out the popular literature, the economics and game theory literature.

Yes, this part of the story belongs in the article. Most readers will be interested in the problem itself. And if you want to get conditional probability ideas across to ordinary folk, you are treading a mine-field. So my suggestion is: be constructive. Present appealing solutions as clearly as possible. Including "conditional" ones. Don't ram fatwa's down people's quotes.

Finally, again, you create a completely false distinction between "strategy" solutions and "conditional" solutions.Conditional probability is a tool to determine a strategy! The reason to do it that way, is because it guarantees an optimal strategy! But you can also guess a strategy and prove its optimality later, if you are so inclined. I hope you agree that the player knows the rules of the game before he goes on the show.

It seems to me that you want the wikipedia page to focus on the variant of the MHP in which the player has never seen the show before and knows nothing about it in advance. He made his way to the final round and chose Door 1. Monty then opened Door 3 and reveals a goat and says "Would you like to switch? By the way, I hid the car myself and I know where it is. Whatever door you chose, I was going to open a door and reveal a goat and offer you a switch. Look, it's written in my contrac, that I have to do it that way today!" .

Weird variant. Richard Gill (talk) 06:48, 11 July 2011 (UTC)

The details about why this assumption must be part of the standard problem certainly belong in a later section of the article, but this assumption is as much a part of the standard problem as the assumption that the host must open a door and must make the offer to switch. -- Rick Block (talk) 16:50, 10 July 2011 (UTC)

IP User banned for 72 hours

The IP addresses 166.216.194.56, 166.216.194.38 and 166.216.194.165 have been banned for 72 hours. See Wikipedia:Sockpuppet investigations/Glkanter/Archive for details. Guy Macon (talk) 20:54, 10 July 2011 (UTC)

Is there still a consensus?

Twice in the past there has been a clearly expressed consensus to resolve the continual argument about this article by adopting my propose structure yet a minority of editors have managed to prevent this structure from being used.

For those new to the article I repeat it here with some clarification

The section titles are indicative of what should go in the sections. If they are considered too POV I would be happy to change them.

1 The problem

Just Whitaker's statement. It is by far the most well-known problem statement. Although it is very vague, most people seem to understand what it is about

2 Vos Savant's and other simple solutions

Without health warnings. These are off-putting for the general audience and unnecessary because all the relevant points will be covered later.

2.1 Vos Savants solutions
2.2 Other simple solutions
2.3 Media furore

This was what made the problem so notable.

2.4 Aids to understanding

Mainly why it matters what the host knows and helping people to understand the solution and any ways of helping people to see the right answer

2.5 Sources of confusion and the psychological aspects
Why people find this problem so hard.

3 Academic criticism of the simple solutions
3.1 Morgan's paper
3.1.1 K&W formulation

4 More detailed and comprehensive solutions

Including the 'conditional' solutions.

3.2 Other 'probability' sources
3.3 Criticism of the criticism
3.4 Summary of 'The Truth' (essentially that there is no such thing in this case, as per reliable secondary sources)

4 Variants

4.1 Other host behaviors
4.2 N doors
4.3 Quantum version

5 History

Rationale

My rational for this structure is:

• No solution or angle on the problem need be omitted.
• All POVs (as expressed in reliable sources) on the problem are clearly and openly presented. It does not push the simple solutions as the only ones. Weaknesses in the simple solutions are made clear at the appropriate point.
• It is the same format as most good text books and encyclopedia articles - easy first then hard.
• Is is accessible to all levels. The general public can read the simple solutions and then bale out. Experts will quickly skip through the simple bits to see if we have covered all the angles properly.
• It promotes cooperative editing because editors can work on the bits that interest them without having to discredit other editors or sections.
• It has been accepted as the consensus twice before.

I am proposing this one last time. If there is no consensus to adopt then I will leave the page, as many other have done, to a fate of eternal bickering. If there is a consensus then we should implement it immediately and keep the article that way until there is a clearly expressed consensus to change it.

There is some scope for modification but I the items in bold are an essential part of my plan to stabilise the article and avoid conflict. Please indicate your support or otherwise below. Martin Hogbin (talk) 16:26, 10 July 2011 (UTC)

Support

Martin Hogbin (talk) 16:26, 10 July 2011 (UTC)

Richard Gill (talk) 07:23, 11 July 2011 (UTC)

Supporting, with Special Opinion - see below. [Machtindex]

Oppose

• Oppose. Separating the so-called simple and comprehensive solutions (especially with intervening material relevant to both) is a disservice to the reader. There's a clear path to give the reader: run the simulation hundreds or thousands of times, and {2/3, switch} is the winner (even Monty agrees). Give the reader a chance to recognize that any intution about 50-50 has a problem. (Why do so many misinformed editors keep inserting solutions showing {1/2, doesn't matter}?) Then provide {explanations, solutions, demonstrations, what-have-you} that explain why switching works. And that should be all WP:DUE solutions: simple, conditional, game theory. The explanations should be readable and offer insight; sadly, many current explanations/solutions do not. The stated/previous structure is a poor compromise that is more about appeasing the two nominal sides of this debate rather than explaining the MHP to the reader. Glrx (talk) 17:50, 11 July 2011 (UTC)

It seems Girx that you think that what is wrong with Martin's proposal is that it does not address the real problem, which according to you is not the structure of the article, but the poor quality of the explanations of the more complex solutions. So, how could we make the more complex more accessible and appealing to the ordinary reader? Well, that is exactly what I have been arguing for ever since I started working here.

Give "easy" versions of the conditional solutions. Give solutions of the conditional problem which use the simple (unconditional) solution as a bridge. Remove the negativity of "so and so said that the simple solutions are *wrong* and you *have* to do it this way". Just present the solution itself. If you want the general reader to learn about conditional probability you have to take a different strategy! Make the conditional solutions appealing and show that though it costs a bit more work, it also delivers a bit more information.

We are in a better state to do this today, than a year or two ago. There are new "reliable sources" which provide the material; previously they did not exist. So all editors should now go home and "digest" the new sources. Come back here again, when they have brought their knowledge of MHP up to state-of-the-art level.

No need to quarrel now about the structure of the article. Better to improve the separate sections "in situ" and only after that discuss again whether or not a new structure is needed. Richard Gill (talk) 17:40, 17 July 2011 (UTC)

I agree in part. Although there are two nominal camps, there is some broad agreement between them. Neither camp is dismissive of the other. Martin's proposal attempts to service both camps, but it awkwardly separates simple and conditional methods with other material. My goals are different from the two camps: I want clear and simple explanations of how to solve the MHP. My target audience is high school students. To that end, your desire to avoid devisive labels and for "easy" versions of conditional solutions is apt. Ultimately, a variation of Martin's proposal may be appropriate -- but it would involve conditional methods being applied to nonsimple variations.

I disagree with some of your other points, but that's for another day.

The existing explanations of methods need work. The Bayes section bothers me the most because it omits the payoff. Instead of an explanation, it's a plug-in-the-numbers proof with little insight. It doesn't even explain that leaving C and S as variables covers all the cases (and triggers a complicated enumeration that MvS avoided but the DT included). I'd like to see a simple explanation of what Bayes' Theorem does with a simple recitation of Bayes' formula. Then I want an application of that theorem where P{Monty reveals a goat} = 1 -- meaning the probability that the car is behind the initial selection is still 1/3. That insight is the payoff, and I don't want it buried in a huge calculation. The more involved proof can follow. At least, that is where I want to go, but I haven't had the time to write it.

Glrx (talk) 19:50, 17 July 2011 (UTC)

Neutral

As always, I am neutral on the content dispute. If I see evidence of a clear consensus (I have not seen this so far despite various claims that a consensus exists) I will support that consensus. Guy Macon (talk) 07:17, 11 July 2011 (UTC)

Discussion

• The claim that this outline represents "consensus" based on past polls is simply incorrect. Polls do not determine consensus. Per wp:poll: "Remember that Wikipedia is not a democracy; even when polls appear to be "votes," most decisions on Wikipedia are made on the basis on consensus, not on vote-counting or majority rule. In summary, polling is not a substitute for discussion." (emphasis in original) -- Rick Block (talk) 16:58, 10 July 2011 (UTC)

In this case we have been discussing the subject for over two years and not one single person has changed their opinion. A majority is the only kind of consensus we are going to get. Martin Hogbin (talk) 17:25, 10 July 2011 (UTC)

Speak for yourself. Perhaps since I still don't agree with you, you haven't noticed that I've moved from a stance that the (well-sourced) criticism stating that the "simple solutions" address the wrong problem needs to accompany the first mention of these solutions (i.e. "health warnings" are required) to a stance that the article should initially present both simple and conditional solutions in a perfectly neutral fashion without favoring either one (i.e. no health warnings, but no favoritism shown to either simple or conditional solutions). You have apparently not changed your position, which I'll paraphrase as "the simple solutions are the correct way to view the problem, anything else should be presented subservient to these solutions and must not even be mentioned until later in the article because we must first convince the reader these solutions are the one, true way to approach the problem". I am seeking a middle ground between Richard's (a) above (my original stance) and his (c) (your original and apparently still current stance). This middle ground constitutes NPOV - between the opposing POVs of "only conditional solutions are correct (and simple solutions address the wrong problem)" and "simple solutions are correct and sufficient (and conditional solutions are of academic interest only)". -- Rick Block (talk) 18:06, 10 July 2011 (UTC)

I find it odd that Martin was unable or unwilling to work towards a clear statement of his position when I asked so I could take this to content dispute resolution,, and yet is now willing and able to run a straw poll containing what appears to be a clear statement of his position. Now he says "If there is no consensus to adopt then I will leave the page, as many other have done, to a fate of eternal bickering.", yet the eternal bickering is a direct result of his and others failure to to work towards a clear statement of the positions Guy Macon (talk) 07:17, 11 July 2011 (UTC)

Why don't we take to Content Resolution the question whether or not we should organize the article according to Martin's proposal? The two names of positions are then "Martinist" and "Anti-Martinist". The outcome is that one side or the other wins. In defence of Martin, you yourself, Guy, gave Martin the lead in re-forming the page according to that proposal, according to the declared plan that afterwards someone else (Rick?) could take the lead in creating an alternative. Only then, with two alternatives side-by-side, we would decide to go for one or the other.

In case of a Martinist victory, everyone can back get to work on the article with a clear plan which allows everyone to work hardest on the bits they understand best, giving collegial feedback to the others on the other bits. (I for one will then try to sneak a "conditional" solution into the early part of the paper by writing it in such a simple way and without even using the words "conditional probability" that no one can complain. Esoecially since it'll be reliably sourced.

In the case ofan anti-Martinist victory, everyone can back get to work on the article (except that Martin might quit for a while, which I think would be a pity, but maybe he needs the break anyway). There will be a more elaborate conditional solution in the early part of the article but criticism and counter-criticism of the different approaches will be kept for later.

Either would be fine. Either way, we get to move on. Richard Gill (talk) 07:48, 11 July 2011 (UTC)

That would be agreeable to me. I am agnostic as to what terms are used and what the description of each is, as lond as everyone agrees that their side is properly described (I don't want to go through content dispute resolution, get a ruling, and then have someone claim their position and arguments were not included)

Does anyone object to Richard Gill's proposal above? I need an alternative so that the two can be compared, of course. Guy Macon (talk) 09:25, 11 July 2011 (UTC)

Yes I do. See below.

Firstly, I have considered that I might be part of the problem, which was why I took a break from this discussion to see if all the argument disappeared. There was no sign of a consensus emerging and the article got into a greater muddle so I have come back to have one last attempt at resolving this conflict.

Guymacon and others. I must ask you to accept my proposal as a good faith attempt to end the disagreement here. I have no objection to stating my position, in fact I will do so below, but my proposal above is not an attempt to push my POV by stealth but a way to avoid conflict. Please look at my bullet points above and see how many you actually disagree with.

I do not like the idea of having two versions. Firstly, I must repeat, this is not my opinion vs Rick's opinion, this is a plan to avoid conflict by adopting a totally neutral approach. Either there is a consensus for my plan to avoid conflict or there is not. If there is, let us do it, if not, I give up. If you all really cannot see that the structure above separates 'vos Savant and the simple brainteaser with media furore' from 'an interesting and instructive problem in statistics/probability/philosophy/decision theory', in a way that, avoids endless POV arguments, and is in accordance with WP policy and the writing of a good encyclopedia article for a wide audience then I despair of the WP process. Martin Hogbin (talk) 09:37, 11 July 2011 (UTC)

Martin, I am not just assuming good faith on your part, it is a conclusion based upon abundant evidence. I also don't believe for a second that you are part of the problem. All available evidence suggests that you, Rick, Richard, Gerhard, Mathindex, Girx, etc. all just want to improve the article. This is simply an intractable good-faith difference of opinion concerning content (AKA a content dispute), with nobody at fault. Guy Macon (talk) 17:40, 11 July 2011 (UTC)

another digression

Special conditions of Machtindex I agree basically with the excellent suggested structure. However, I find it VERY incomplete. The recent viewpoint on the MHP as THE MONDEE GILLS GAME (MGG) embeds the MHP in combinatorial games. In the classical Whitaker/vos Savant setting this is the most primitive version of a hide-and-seek game played on a 3 x 4 playboard with four chips (prize, contestant,checked door, contestant) which is solved by dominance. The game has complexity level one point higher than tic-tac-toe. All difficulties connected with conditional probabilities, and with probabilities at all disappear at once by this approach. To give you idea what I am talking about: think of 3x3 tic-tac-toe, where the first player makes a move |x|-| and then covers two fields by matchboxes, so that the second player does not see where x stays (say, on the second player's first move). The MHP-MGG is much of the same level of difficulty, albeit with nice "switching combinatorics". Once the problem has a combinatorial solution by the dominance, which in the MHP-MGG instance is there, all probability solutions with arbitrary asymmetric priors and arbitrary payoffs for finding the prize at door a,b,c for premium A,B,C, whichever this A,B,C is probability law, or money rewards, or smth else are all banal. The probabilistic interpretation of A,B,C is posted on the arxiv.org. THE DOORS is downloadable from there. Would you come up with idea of playing tick-tack-toe by making random legal moves? My constructive suggestion is to include The Mondee Gills Game as a subsection in the most elementary part of the story. And in more complex part with antagonism of seeking and hiding players. At present there are presentations undesrandable by 5 year old kids who are capable of understanding the rules of the game: choose door, then hold or switch. What Host is doing is completely irrelevant, he is a dummy player, which only makes the situation more interesting. The Bayesian "machinery" is an excellent way to exercise the conditional probability However, it is beaten completely by the mentioned combinatorial approach. The 20 yrs-dispute conditional-vs-unconditional is a SHAME for the probability community. Leaving big traces of it on the Wiki page only says how isolated are some probabilists from other mathematical developments, in the first turn in computer science, game theory, machine learning, operations research and combinatorial optimisation. The Mondee Gills Game as a 2-person 0-sum was solved by Olle Haggstrom in his textbook Streifzuege... translated from Swedish. Complete game-theoretic discussion with tree presentation and information sets is available: THE MONTY HALL PROBLEM IN THE GAME THEORY CLASS for your downloads from arxiv.org The informational aspect of the problem (Revealer Host) is addressed in the case of four doors, and posted on the arxive.org. You may download THE UNLUCKY DOOR. The latest write-up of THE MONDEE GILLS GAME: STRATEGIC DOMINANCE FOR DUMMIES is available upon request. The first yet unpublished version of THE MONDEE GILLS GAME with discussion of the controversy is available upon request. The earlier version of the work THE MONTY HALL PROBLEM: SWITCHING IS FORCED BY THE STRATEGIC THINKING is available from arxiv.org. These are the latest. Add Richard's Stat Ned paper, add Olle Haggstrom's book. Many Wiki articles are based on 3-4 publications, ckeck the article Robbins' problem. Look how Benford's Law page is organised: nobody struggles there on the occasion of much more notorious phemomenon of the first digit. And everybody who wants to promote knowledge has their space. The basics of the game-theoretic, optimisational and combinatorial approaches to the MHP HAVE BEEN CREATED. Only blind cannot see this, sticking in 20yrs old nonsense. If we are here to bring knowledge to laymen instead scholastics then revert the moves in the discussion first SWITCH/HOLD the door then OPEN the door. Trust me: the moves are exchangeable: this is a theorem which is equivalent to the optimality of always switching.Machtindex (talk) 08:58, 11 July 2011 (UTC) I love the MGG approach but it has to become part of the mainstream literature on MHP before wikipedia can incorporate it in a big way in an article on MHP. You really must carefully study what wikipedia is! See WP:ABOUT and WP:FIVE. Please also try to learn how to do wikipedia markup. Richard Gill (talk) 10:19, 12 July 2011 (UTC)

Martin's POV

I am sure you all know this already but I have been asked to state my POV. My POV is that, taking into account the original context and the intended audience, the simple solutions to the MHP are perfectly correct. The 'conditional solution' is a minor academic diversion (since retracted by its originators) of interest to neither the general public or real experts in the subject, but of some value in teaching elementary conditional probability theory. Martin Hogbin (talk) 09:44, 11 July 2011 (UTC)

Conclusion by Martin

There is a small majority for my structure but not enough that I could claim a consensus. I guess this argument will now continue indefinitely, or at least until enough new editors join to produce a consensus. This will not help the article improve. Martin Hogbin (talk) 09:16, 17 July 2011 (UTC)

Or you could try my solution -- Going through the process of Wikipedia Content Dispute Resolution. Wikipedia has a process for resolving this sort of content dispute. Why not give it a chance? Guy Macon (talk) 21:38, 17 July 2011 (UTC)

Why is the MHP said to be a probability puzzle, and not just the question to switch or not

To Martin Hogbin and everybody in the forum. Could you please answer one simple question: Why the MHP is a probability puzzle, and not just the question to switch or not. If somebody finds the query strange, think of the chess. Playing white or black pieces is decided by tossing a coin. Should we appeal to the probability theory to find winning strategies?RocksAndStones (talk) 15:48, 18 July 2011 (UTC)

Good question, RockAndStones, see The Monty Hall Problem is not a Probability Puzzle (It's a challenge in mathematical modelling) or The holy grail of Monty Hall studies, saying: ". . . Therefore the 2/3 success-chance of always switching can't be beaten. I would call this a proof by coupling."

But, aware of two doors having double chance, some said that you "have to consider that – as to the actual game – some slight additional information could be revealed by some conjecturable special method of the host in opening one door whenever he got two goats to show. Thereby, in each and every game he could be signaling additional info, e.g. that actually the odds on the host's second still closed door are "1" (and switching will win for sure in 1 out of 3 cases), or that the odds could be at least 1/2 likewise (in two out of three cases), according to their casual technical assignment. And of course they are free hereupon to do a lot of unprofitable conditional probability calculus, but that never will nor can proof that staying could ever be better than to switch. They misunderstood the question "to switch or not to switch", utterly ignoring that the 2/3 success-chance of always switching just can't be beaten. Gerhardvalentin (talk) 16:45, 18 July 2011 (UTC)

Added: One Auto, Two Goats  –  And they (Morgan et al.) even said that it is incorrect to reason: "As the car originally is equally likely to be behind each of the three doors, the combined chance of doors 2 and 3 altogether to hide the car is 2/3". They say this is a false reasoning, because "after the host has opened door #3 it is evident that the likeliness of the car to be behind door #3 was not 1/3, but it was zero":

That "AGG, GAG, GGA, each point having probability 1/3" is not a solution to the stated conditional problem is apparent in that the outcome GGA is not in the conditional sample space, since door 3 has been revealed as hiding a goat." – That's what "they" say, all of that completely irrelevant to the decision asked for, and completely irrelevant to the given fact that "the switch here and now-decision" never can be beaten by any other decision. Gerhardvalentin (talk) 08:37, 19 July 2011 (UTC)

RockAndStones, is there reason you have asked me in particular?

I believe that it is important to point out that not only should you switch but that your probability of winning doubles if you do so. I am not sure if that answers your question. Martin Hogbin (talk) 17:30, 18 July 2011 (UTC)

The question is whether you should switch or not. The answer is based on whether or not the probability the car is behind door 1 is the same as the probability the car is behind door 2 (after the host opens door 3). Answering the question based on the probability means the problem is a probability puzzle. As it turns out (with the usual interpretation of the problem), the probability the car is behind door 2 (which naively appears to be the same as the probability it is behind door 1) is actually twice the probability it is behind door 1 - so you should switch.

There are both direct and indirect ways to see this.

An indirect way is to imagine what would happen on (say) 300 shows if no player switched (about 100, or 1/3, of these players would win a car) vs. if all players switched (if 100 win by not switching, then 200, i.e. 2/3, must win by switching - all 100 players who would have won a car by not switching lose, but all 200 players who would have lost by not switching win if they switch). Note that this includes players who initially pick door 1 who switch to either door 2 or door 3 (depending on which door the host opens) - but if 2/3 of the players who pick door 1 and switch to either door win, then 2/3 of the players who switch to each door should as well (everything else being equal).

A direct way is to compute the conditional probability that the car is behind door 2 given that the player initially picked door 1 and the host has opened door 3. The host must open door 3 if the car is behind door 2 (this happens 1/3 of the time) but chooses whether to open door 2 or door 3 if the car is behind door 1 - assuming this is an even choice then the host is opening door 3 when the car is behind door 1 1/6 of the time, i.e. if you pick door 1 and see the host open door 3 you have a 2-1 advantage by switching (the conditional probability the car is behind door 2 is 1/3 / (1/3 + 1/6) ). -- Rick Block (talk) 18:41, 18 July 2011 (UTC)

MHP - thought of as a decision problem - can be completely solved without any probability. A player chooses a door and will switch or not switch when the host opens either of the two other doors. For instance: "choose door 2, hold if door 1 is opened, switch if door 3 is opened" (there are 3x2x2 possible strategies - 3 initial choices, hold or switch when the lowest numbered of the other two is opened, hold or switch when the highest numbered of the other two is opened). To any strategy which involves "hold" in some situation there is a strategy of always switching which does at least as well, wherever the car is, and whatever the host does. For instance, in my little example: with the given strategy, if the car actually were behind door 3, the host would be forced to open door 1, the player would hold, and not get the car. But the player who used the strategy "choose door 3 and switch, whatever the host does" also doesn't get the car if it's behind door 3 but does get it in every other situation, whatever the host does.

So, a little bit of thought in advance shows that whatever one should do, it makes no sense ever to "hold". The only question is "which door to choose first".

People who are interested in probability can go on and say more if they want to, but if you just want to know what action to take, the answer is "switch". References: recent articles on arXiv.org by A.V. Gnedin. Approach: using the notion of dominance from game theory.

If you want to add some probability to all this, one might for instance be prepared to assume that the car is initially equally likely behind every door. We see that however one chooses an initial door, if thereafter one switches one gets the car with probability 2/3. This can't possibly be improved since any strategy which sometimes plays "hold" is case-by case worse than an always switch strategy. Consequently, every conditional probability that the car is behind the other door given your first chosen door and the door opened by the host must be at least 1/2. Nothing was assumed here about how the host picks his door to open. If we are also given that he does this with equal chances either door, when he has a choice, then the whole problem is symmetric and the chance of winning by switching can't depend on door numbers. The conditional probability of winning by switching is trivially equal to 2/3 since winning by switching is statistically independent of the door numbers one observes in any particular case (by symmetry).

All in all, one needs no probability at all to solve the decision problem, and after that, if one is interested in probabilities, and if probabilities are available - because more information has been given - then one can get all the probabilities one likes without calculating conditional probabilities at all.

Unfortunately it will be 10 years before this approach is in all the standard text books, so in the meantime wikipedia readers are going to have to be told the difficult ways to solve the problem, and the article will be at least 20 times longer and 20 times less accessible than is necessary. Richard Gill (talk) 20:02, 23 July 2011 (UTC)

Would it help to disambiguate "the" question?

It is my perspective that there are two distinct interpretations of what the question is asking. (This is different from the binary "switch or not" vs. the quantitative "what are the odds" distinction.). These have sometimes been described as (a) asking for a strategy independent of the door opened, and (b) asking for a posterior decision given an open door. More fundamentally, if somewhat abstractly, I think of these as stipulating that (a) "say, door #3" means treat the doors as undistinguished, and (b) "say, door #3" means either door might as well be distinguished. The problem, as I see it, is that some solutions to "the" MHP can be considered superfluous digressions for answering MHP(a) or insufficient for answering MHP(b) without justifications or "health warnings".

My preference would be to distinguish at the outset, nonjudgmentally, between two interpretations of the question, rather than treating solutions as different opinions about the answer to life, the universe, and everything. Advantages of this approach could be (1) presenting simple solutions as unreservedly good answers to a simple question, (2) shedding light on why conditional analysis is important, for the benefit of readers who are unfamiliar with it, and (3) mitigating disputation by reducing occasions for criticizing or justifying solutions. Disadvantages of this approach could be (1) the distinction may be too subtle for some readers to follow, (2) the relevance of some material does not fall exclusively under a single interpretation, and (3) exacerbating disputation by bifurcating treatment of "the" MHP.

Is there any chance this approach could lead to a way forward? ~ Ningauble (talk) 18:11, 10 July 2011 (UTC)

In my opinion, it definitely could lead to a way forward. In particular, it seems extremely promising as a way of getting to the short name and descriptions that are needed before this can be submitted to Content Dispute Resolution. Good thinking! Guy Macon (talk) 20:54, 10 July 2011 (UTC)

I agree that there are two interpretations of "say Door 1". Vos Savant even confirmed her intention: the numbers are meant from the outset to be irrelevant. In fact Whitaker's actual words, which were revealed in Morgan's response to Martin's correction note last year, did not specify any numbers but explicitly talked about "switching" or "not switching" in general.

But I disagree that one can distinguish between looking for (a) a strategy, (b) a posterior decision. Under the notion "strategy" is included strategies which could involve specific door numbers. And looking at posterior probabilities - which one can perfectly well do in advance - is a way to fix a strategy. In fact, the good reason to investigate the conditional probabilities iis because it guarantees that you'll find an optimal strategy this way. However if you are happy enough with the incredible gain of 2/3 overall as compared to 1/3 overall, then there is no reason to hurt your brain trying to figure out if you could do better still. Alternatively, come up with a simple reason why 2/3 overall can't be beaten, and then you needn't look at conditional probabilities at all. Theorem: A constant strategy is optimal if and only if the conditional probability favors this strategy in each individual case.

It is such a pity that the standard probability and statistics literature never says why it's a good idea to look at conditional probability. It's even sadder that the more unthinking writers even say you must. Richard Gill (talk) 07:17, 11 July 2011 (UTC)

In exemplifying that some have characterized a difference between strategy and posterior decision (I might also have mentioned a difference between unconditional and conditional) I did not mean to define the difference I am raising thusly, but to contrast with what I believe is a more fundamental and useful difference in ways of reading the question, as distinct from ways of approaching an answer. There is a real difference between stipulating undistinguished doors and concluding, when considering distinguished doors, that they might as well be considered undistinguished.

I think your remarks about strategy support my view that some of the dichotomies that have characterized disputes here are inapt. I am suggesting differentiating between (a) a question about undistinguished doors and (b) a question about distinguished doors. Do you think this would be a useful way to present (a) simple solutions to a simple problem and (b) deeper analysis of a subtler problem (some of which, interestingly, points back to the simpler problem)? ~ Ningauble (talk) 13:10, 11 July 2011 (UTC)

The MHP was initially intended as a simple puzzle. There is a long-standing convention in simple puzzles that all necessary assumptions are made in order to keep the puzzle simple. Thus "say, door #3" should be taken to mean 'another door'. We know, in fact, that this was exactly what vS meant when she added it. Everything else is an academic diversion, interesting in its own right, but not in the spirit of a simple puzzle. Neither is it in accordance with the principle expounded by Seymann that when an expert answers a question from a layman they should strive to find out what the questioner actually wants to know, rather than stick strictly to the exact wording.

You can always add complications to a problem if you try hard enough, and there is no reason that we should not follow those sources which have extended or dissected the MHP but please, after the simple puzzle has been simply resolved. Martin Hogbin (talk) 16:14, 11 July 2011 (UTC)

Notwithstanding that my own preferred reading of the problem is MHP(a) (i.e. that "say, door #3" not only means some other door, but stipulates that the choice makes no difference), I must still acknowledge there is a substantial body of literature that addresses MHP(b). I think that distinguishing between these two readings of the problem would help to delineate between the part of the article that is of most interest for readers who are curious about the "simple puzzle", as you call it, from the part of the article that is of interest for readers who would like to explore further mathematical considerations. Furthermore, I think it more than satisfies the aim of WP:TECHNICAL because it does not treat "simple" solutions as mere glosses, but treats them neutrally and objectively as real solutions to a simple riddle.

Do you think that differentiating between these two interpretations of the problem could facilitate your aim of treating "academic diversions", as you call them, after the simple puzzle has been simply resolved, by delineating where one leaves off and the other begins? ~ Ningauble (talk) 17:38, 11 July 2011 (UTC)

Yes, of course we must distinguish between what you call MHP(a) and MHP(b) but in the correct place, which is after the simple solutions have been given. Many readers will have no interest at all in the difference between the two problem, especially as (for the standard problem) they both give the same result. There are several rationales for giving the simple solutions (without health warnings). One is, as you state, that the question can be taken to mean MHP(a) (this is why we should not give the K&W problem description at this stage) another is that even for MHP(b) the simple solutions are perfectly correct by the application of a simple, obvious, and intuitive symmetry. In the symmetrical case the conditional probability must be equal to the unconditional one by the law of total probability. Falk refers to this logic as impeccable. Although we do not state the symmetry, my guess is that most readers intuitively assume this. Just ask anyone whether they think it makes any difference whether the host opens door 2 or door 3.

My suggestion is therefore, first give the simple solutions to the puzzle (without let or hindrance) then cover all the rest of the MHP openly and clearly according to reliable sources and including the criticism of the simple solutions. Why would anyone not want to do that? Martin Hogbin (talk) 18:06, 11 July 2011 (UTC)

Well, Rick has explained many times why he thinks this creates a bias. I believe that at the moment he is in a minority but the fact remains this is *the* dispute which has been going on for years. First we had mediation (2x) without success, then arbitration, which led to the removal of two editors, a welcome influx of new ones, and the article is "under probation". We now have a "probation officer" (Guy Macon) and everyone is doing their best to behave in a civilized way. In my opinion Guy is doing a splendid job. Now, he proposes we move to "content dispute resolution", but first we would have to agree what *is* the content of the content dispute. So my question to Guy is, again: can't we take Martin's proposed framework as the issue of content dispute, so that it can be settled one way or another, whether or not we follow this proposal? After further wrangling, involvement of yet other new editors, there will be an outcome. Either Martin or Rick will be highly disappointed with it. But at least we get closure on this question, for a while. Then everyone who wants to can concentrate on constructive improvement of the article within the constraints of the framework proposed by Martin or of one proposed by Rick.

BTW, I would vote for Martin's outline. I think he gives cogent reasons why this is an apropriate way to structure all this material on wikipedia. He explains clearly why it *is* a compromise aimed at achieving broad consensus. In particular, it does not actually strongly favour his own personal point of view on MHP.

Ningauble, I'm glad that you agree about the inaptness of some dichotomies which have been floated here. And I believe that the distinction between your MHP(a) and MHP(b) is actually one of the deeper reasons why many ordinary people find a conditional approach a meaningless diversion. In the preprocessing step (before proceeding to formal reasoning) they simplify to problem (a), whether following cues in Marilyn's wording or following their own instinct. From a subjectivist point of view, their instinctive move is justified. One can just as well argue beforehand, as afterwards, that door numbers can be neglected. See e.g. Georgii for a text book presentation following MHP(a), see my notes on my university home page. Unfortunately, this observation is not prominent in the literature. Richard Gill (talk) 10:51, 12 July 2011 (UTC)

To distinguish "did the host open door 2 or door 3" refers to some suspected underlying asymmetry, for in the asymmetric case the odds (given the host opened door 2 and given he opened door 3) could slightly differ from 2/3, nevertheless switching will still have an overall 2/3 probability. But as long as such asymmetry and its direction is not known for sure at the outset but can only be considered as suspected to exist for this one game given, it makes no sense to use it, especially as it can't change the decision to switch. It remains what it is, a less important side aspect that could be mentioned also, later on. What the sources say: In short, as long as you don't know for sure about existing asymmetry, to distinguish between door 2 and door 3 is not worth the trouble. But I would like to see a conditional formulation, clear and in odds form, already in the beginning also, just as an aid to understand that the odds are not 1/2 in general, but that in general the odds are 2/3. Gerhardvalentin (talk) 15:38, 12 July 2011 (UTC)

Gerhard, if you don't know for sure about existing asymmetry, in particular if your knowlege about possible asymmetry is neutral to the direction of asymmetry, then for a subjectivist (probability represents your state of knowledge) the host is equally likely to open either door when he has a choice. For such a subjectivist the problem is symmetric and hence by symmetry the door numbers can be thrown away from the start. That is what Georgii does in his book. A frequentist has a harder life. The sources do not say what you say they say. They do not say "to distinguish between door 2 and door 3 is not worth the trouble". Richard Gill (talk) 09:28, 14 July 2011 (UTC)

Right, but that was just "in short" what they say (being not just textbooks to teach conditional probability theory). Reliable sources on MHP say:
The actual numbers of door chosen and door opened are irrelevant to deciding whether to switch or stay.  –  Or:
Simple plus symmetry: by symmetry the probability that the car is behind door 1 cannot depend on whether the host opened door 2 or door 3. The unconditional probability was 1/3. Therefore the two conditional probabilities are equal to 1/3 too. Reference: Bell (1982).
And other sources on the MHP that don't even pay regard to whether #2 or #3 was opened, leaving associated unecessary assumptions aside. Gerhardvalentin (talk) 13:44, 14 July 2011 (UTC)

I do not mean to make a big deal of indistinguishable doors in a way that implies simple solutions are only relevant to one reading of the problem. Notwithstanding the many and various reasons one may interpret, assume, hypothesize, or conclude that the doors are undistinguished, or choose not to, the fact of the matter is that this choice characterizes a sharp and fundamental distinction between different ways of expressing solutions, and underlies much of the contention over correctness of solutions. I believe it would greatly clarify the article for the widest possible audience to explicitly indicate this difference rather than meandering from one solution to another without providing this context.

What I would like to see is a main section on simple solutions prefaced by a very brief statement of what is being stipulated or assumed (not a health warning). Concluding the section by observing that considering distinguishable doors leads to the same conclusion, and that some approaches treat it as a distinction that does not make a difference for reasoning to reach that result, would be a good segue into subsequent sections on less "simple" treatments.

Honestly, I think it would make the article clearer; and I do not think this widely noted distinction is original research, nor that using it to provide context creates structural bias for or against different approaches to the problem. ~ Ningauble (talk) 15:18, 13 July 2011 (UTC)

I agree with your conclusions, Ningauble. I also agree that simple solutions are not *only* relevant within one particular reading of the problem. A glance at the sources shows that this is not true! Improving your overall success-rate from 2/3 to 1/3 is a good reason to switch, however you read the problem. And it only requires the assumption that your initial choice has chance 1/3 to hit the car. Richard Gill (talk) 09:28, 14 July 2011 (UTC)

I agree too. The simple solutions have many diverse justifications, however, that is not the main reason we should have them first. The main reasons that we should do this are firstly, they are simple, and secondly, they are by far the most well-know and notable. The fact that the simple solutions do have many justifications makes the decision to put them first an easy one.

Once you go beyond the simple solutions and look at the philosophy of the subject and different possible interpretations you lose the interest of many readers. This is a separate subject which in my opinion deserves to be treated properly and in depth. Attempting to give a half-baked analysis of the philosophy of this problem at the start is a waste of time for everyone.

The only correct solution to any probability or statistics problem is the one that provides the questioner with the information that they wished to acquire. Martin Hogbin (talk) 09:13, 17 July 2011 (UTC)

To summarize responses thus far (If I have misunderstood anyone's position the fault is entirely mine, and I welcome the opportunity to be corrected.) —

• Guy believes the idea has merit, but considers it a matter of dispute. (Branding my suggestion as disputatious is not unreasonable: I broached it with some trepidation that I might be pointing out the elephant in the room.)
• Richard and Martin, after some clarification, are agreeable to the idea. (The amount of clarification needed probably indicates that it would take considerable effort to find suitable wording and implement it in the article.)
• Gerhard prefers to have an explicitly conditional formulation included in the beginning, and notes that some sources leave associated unnecessary assumptions aside as irrelevant.

Before trying to draft specific language, I would appreciate more feedback on this idea from other folks who still watch this article. I don't want to proceed with a draft that fails to take other perspectives into account, or to spend time on details if there is little chance this approach could lead to a way forward. Thanks. ~ Ningauble (talk) 16:23, 22 July 2011 (UTC)

What source or sources are you intending to use for the proposed discussion? As I see it, the main problem is that the sources presenting simple solutions essentially never say what problem they're exactly addressing (a or b). One might assume they're addressing (a), but since these sources don't make it clear it becomes vanishingly close to wp:synthesis for the article to make this clear on their behalf. As I've said repeatedly, there are plenty of sources that distinguish (a) and (b) nearly all of which say in one way or another that vos Savant's wording of the question implies (b). Rather than attempt to make this distinction early on, I think a more neutral approach is to present both a simple solution and an entirely accessible conditional solution as two different approaches without trying to favor either one. Both of these approaches are extremely common and (IMO) both should be presented very early on in the article. The discussion of the subtle differences between these approaches can come later, but IMO it's distinctly not NPOV to present the simple solutions at length without any mention whatsoever of the equally common conditional approach (particularly since there are numerous sources that explicitly prefer the conditional approach). Presenting both, as equally valid approaches, lets those readers who are thinking of the "unconditional problem" see a simple solution (these readers will presumably gloss over the conditional explanation) and but also lets those readers who are "misled" by the problem statement into thinking of the door 1/door 3 (conditional) case see a solution that pertains specifically to this case. Presenting them as complementary, not antagonist, approaches is IMO more NPOV and likely to be more convincing as well. -- Rick Block (talk) 19:45, 22 July 2011 (UTC)

Eliminate anything that is completely irrelevant to the decision asked for, but is just suitable for the mathematics class-room. I prefer to show the Bayes' rule in odds form just at the beginning also, easy to read and understand to anyone. Show the remarkable and controversial history later. Cut out the conditional solutions part of the page and have a link instead to where it belongs, to Bayes' theorem. Most of all: the article is about the famous paradox and the relevant decision, not just a lesson in conditional probability theory only. Gerhardvalentin (talk) 09:05, 23 July 2011 (UTC)

Rick asked "what sources?" I have told him so many times ... The text-book by Georgii, the encyopedia article on www.StatProb.com by one R.D. Gill. StatProb is hosted by Springer and published by a consortium of all the major national and international societies for statistics and probability, both professional and academic. Read the sources! Read the latest authoritative tertiary literature! Richard Gill (talk) 18:59, 23 July 2011 (UTC)

The question was for Ninguable, who seemingly expresses concern about how to cite this discussion in the very beginning of this thread. I have read the textbook by Georgii and your encyclopedia article on StatProb - and the original paper by Morgan et al, and Gillman's column, and the textbook by Grinstead and Snell, and the paper (and book) by Falk, and the paper by Puza et al., and the paper by Eisenhauer, and the paper by Carlton, and the paper by Rosenthal, and the paper by Lucas et al (and many, many more). There is no question that some authors who know what they're talking about approach the problem by examining the result of a predetermined strategy to switch. The question is whether the vast majority of popular authors who present "simple" solutions claim to be saying that the probability of winning by switching to door #2 given you've initially chosen door #1 and have seen the host open door #3 is 2/3 (with the decision point being standing in front of a closed door #1 and closed door #2 and an open door #3 showing a goat, i.e. after the host opens door #3), or whether they're only claiming with a predetermined strategy of switching the probability of winning the car is 2/3. The usual problem description makes it clear we are to consider the door 1/door 3 case but the usual simple solution ignores this case (and instead considers only predetermined strategies) - and it is this "bait and switch" that makes the simple solutions so hard for most people to swallow. As Eisenhauer puts it "what could and should have been a correct and enlightening answer to the problem was made unconvincing and misleading". I'd rather not have the Wikipedia article repeat this mistake. -- Rick Block (talk) 20:00, 23 July 2011 (UTC)

Dear Rick,

(1) You seem consistently to misunderstand me and hence misrepresent what I'm saying. I am *not* talking *only* about predetermined strategies. I am talking about *all* strategies. Strategies determined in advance, strategies determined on the fly. Strategies which always recommend "hold" or "switch" independently of the door numbers the player will see, or strategies which allow different actions in different circumstances. Strategies which use coin tosses and strategies which are deterministic. Also included is the strategy "calculate conditional probabilities after the host has opened a door and decide on the basis of that conditional probability". Also included is the strategy "choose door 1 and when a door is opened think about it for a while and make up your mind what to do". Whether the calculation or the thought is done on the fly, both possible calculations (or thoughts) can (for the purposes of argument) be thought of as done in advance. The decision moment makes no difference. The solution I describe tells us what to do in all situations hence also in the situation that 1 was chosen and 3 was opened. The moment in time at which the decision is made is totally irrelevant. One may in advance consider what might possibly happen in the future and what one would then decide. A solution method for the "1,3" case becomes a solution of every case, by permutation of the numbers. Vos Savant said "say", by way of example.

(2) I am not aware of any popular author who presents a simple solution and who moreover claims to be saying "the conditional probability of winning by switching to door #2 conditional on the event that you've initially chosen door #1 and have seen the host open door #3 is 2/3". Of course a section on simple solutions mustn't contain obvious untrue statements, at least, it would not be helpful for the reader to include obviously untrue statements.

(3) I too have read all these sources, and a great deal more. I've corresponded with the authors of several key sources and written a few myself. Richard Gill (talk) 20:22, 23 July 2011 (UTC)

1) I know perfectly well what you're talking about.

2) Most, if not all, popular authors at least implicitly say "the conditional probability of winning by switching to door #2 conditional on the event that you've initially chosen door #1 and have seen the host open door #3 is 2/3". Would you like quotes?

Here's vos Savant The winning odds of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door. To illustrate this, let's say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger. She doesn't use the words "conditional probability" here, but she's clearly saying the conditional probability the pea is underneath the shell you originally chose must be the same as it's prior probability regardless of which shell is turned over.

Vos Savant's three shells are indistinguishable. Her argument is correct for three shells. But anyway, we are not going to copy every solution in the literature. We are going to give generic solutions which are widely found in the literature. And be careful that what we say in every case is both true to the originals and mathematically correct. There is no need to present any arguments which are wrong! Richard Gill (talk) 19:21, 24 July 2011 (UTC)

Without Bayes wrong from the outset? – Rick, Falk says indeed "if this bias exists and you know about that bias". But read the sources: *Whether or not you know about her bias* is completely irrelevant, you will have learned *nothing* to revise your decision to switch to the offered "door #2". Even the greatest difference in "conditional probability" can nor will ever be of disadvantage for switching, and switching now and here gives you the maximum benefit possible. Quite another issue is teaching and learning conditional probability theory in a maths classroom, although without any relevance to the correct decision asked for, and never a sine qua non for the MHP. Show it where it belongs, by a link to Bayes' theorem. Gerhardvalentin (talk) 08:49, 24 July 2011 (UTC)

Not exactly in the same class as "popular" authors, but here's Devlin: Suppose the doors are labeled A, B, and C. Let's assume the contestant initially picks door A. The probability that the prize is behind door A is 1/3. That means that the probability it is behind one of the other two doors (B or C) is 2/3. Monty now opens one of the doors B and C to reveal that there is no prize there. Let's suppose he opens door C. Notice that he can always do this because he knows where the prize is located. (This piece of information is crucial, and is the key to the entire puzzle.) The contestant now has two relevant pieces of information: 1. The probability that the prize is behind door B or C (i.e., not behind door A) is 2/3. 2. The prize is not behind door C. Combining these two pieces of information yields the conclusion that the probability that the prize is behind door B is 2/3. You tell me, is Devlin saying the conditional probability the prize is behind door B is 2/3 or not?

That was Devlin, a mathematician, trying to give a conditional solution, and doing it wrong. As he later admitted, he had skipped a step. It is easy to fix using symmetry. This has got absolutely nothing to do with your argument. It does illustrate however (a) that conditional solutions can be simple, and (b) even great mathematicians can slip up occasionally. So fir God's sake please let's only put correct arguments in the article. There's no point in copying known mistakes, or claims which are not only false but which have been refuted (such as Rosenthal's, or Snell and Grinstead's claims that Whitaker is asking for a conditional probability.) Richard Gill (talk) 19:36, 24 July 2011 (UTC)

3) With all due respect, you are but one of the many thousands of professors of probability and statistics. There is no guarantee whatsoever that your POV reflects the dominant thinking in the field - in fact, since you're actively publishing in this area one might assume you're attempting to pursue novel or otherwise interesting lines of thought that are nearly by definition not the dominant thinking in the field. The point of the article is not to reflect Richard Gill's thoughts about the MHP - but rather to represent "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources". That you apparently disagree with Morgan (and presumably Gillman, and Grinstead and Snell, and Rosenthal, and Eisenhauer, and Falk [who is a psychologist], and others) distinctly does not mean that we can or should ignore the POV these authors (as well as the many, many others who simply present a conditional solution without commenting on "simple" solutions) express. -- Rick Block (talk) 01:35, 24 July 2011 (UTC)

Thanks for your respect. I published some mathematical details which are common knowledge in my field, and which seemed to me potentially useful for editors here since they helped defuse the conflict by offering bridges and clarification and simplification, but which unfortunately seemed not to exist in convenient citable form in print. You would rather ignore all this and instead bother the reader with a conflict which is now past history and of no particular interest to the general reader. Too bad for those who come to Wikipedia hoping for information which is clear and up to date and attractively presented. Richard Gill (talk) 19:49, 24 July 2011 (UTC)

This is in reply to Rick's initial response in this thread, which raises several issues. Pardon the lateness of my reply.

1. In my non-specific request for sources, I actually had one particular question in mind. It is more specific than the matter of some sources treating the doors as distinguishable and others omitting or disregarding such distinctions. To wit: some time ago I briefly searched for sources that address the use of "say" in mathematical discourse to signify that an arbitrary distinction is being introduced for the convenience of concrete exposition but that it is expressly to be understood that the situation is symmetric with respect to the choice being distinguished. Although I have a shelf full of textbooks that exemplify this common usage, I did not find sources that discuss it in reference to the semantics of MHP, and I have not found any source that gives an authoritative definition for this façon de parler. I would be grateful if someone could find such sources.
2. As discussed with Martin and Richard above, it is not my intention to presumptively ascribe reasons why some sources do not distinguish doors, but to identify as "simple" those that do not do so, by initially indicating simply that this may be stipulated or inferred.
3. I very much do want to treat these as equally valid approaches in a complementary presentation. I believe that mixing different approaches together without clarifying this distinguishing characteristic creates just the sort of "he said, she said" back-and-forth that is eschewed by WP:STRUCTURE. (In the current revision of the article,[2] e.g., using the reading indicated in point 1 above, interposing the K&W formulation of the problem between vos Savant's question and answer casts the latter in a prejudicial light because K&W substitute an expressly probabilistic uniform distribution in a temporal context for vos Savant's a priori symmetry used for a combinatorial solution. Some read it differently.)
4. In pursuit of the WP:TECHNICAL approach for putting the most understandable parts of the article up front and writing down a level, what I want to do is replace subjective assessments of which approaches are "simpler" with the objective characteristic of not treating the choice of goats as a distinction that makes a difference. That symmetry is a fundamental measure of simplicity is attested by many sources in the literature of science and mathematics, including foundational ones of historic significance. There should be no need to cite or discuss that literature, but I believe this is a sound, objective principle for organizing the article. It is this symmetry which makes the truly simple solutions almost as intuitive as the 50:50 fallacy.
5. Regarding the accessibility of conditional solutions, I lean toward the view you expressed earlier (much earlier), though you may have intended hyperbole, that "the absolute probability of a layperson understanding a conditional probability analysis is near 0." This is not to deprecate conditional approaches themselves, but to argue that it is better not to include them in the opening sections. Contributors here are exceptionally knowledgeable, and we would do well not to underestimate the mathematical illiteracy of the general public which, at least in the US, is widely documented.
6. Readers can indeed be confused and misled by interpreting the problem in a way that is inconsistent with one or more of the solutions. My intent is to remove that confusion by providing context for the solutions that identifies approaches to the problem which they address.

In the parable of the blind men and an elephant it would be original research to suggest why each person is inspecting the elephant from a different perspective; but the story would be incoherent if it didn't explain that this is exactly what they are doing — that would be ignoring the elephant in the room. ~ Ningauble (talk) 20:21, 24 July 2011 (UTC)

1) Krauss and Wang have a very short discussion of the meaning of "say": "Although, semantically, Door 3 in the standard version is named merely as an example ('Monty Hall opens another door, say, number 3'), most participants take the opening of Door 3 for granted and base their reasoning on this fact." They go on to explain that this assumption (that it is specifically Door 3 that has been opened and not Door 2) makes the intuitive solution (what are called "simple solutions" in this context) inaccessible. Whether "say" is meant as an example or not, the context of the problem (a game show, with 3 numbered doors) makes it contextually clear the doors are distinguishable. The use of "say" might very well imply the solution is meant to be the same regardless of which door the player initially picks and which door the host opens, but this is a subtly different argument than that the doors are indistinguishable (if the doors are indistinguishable then the solutions must be symmetrical, but not necessarily vice versa). Once again, what I'm suggesting is that we discuss this topic later in the article, but present both "simple" and conditional solutions early on.

2) Implying or directly saying a published "simple solution" is based on indistinguishable doors if the source does not say so itself or if there isn't some secondary source that says this is WP:OR.

3) I fail to see how presenting two solutions, both of which say the answer is the probability of winning by switching is 2/3, can create a "he said, she said" back and forth. Again, my suggestion is to present both sorts of solutions while remaining absolutely and completely neutral about whether one is better than the other. Interposing a discussion that says some solutions assume the doors are indistinguishable and that some don't, and providing an "indistinguishable context" for the solutions that follow (without basing this on a specific secondary source) is at least as biased as the current content (which is sourced to a highly reputable source). Per my comment above, since the game show context implies the doors are distinguishable, another interpretation is to justify the simple solutions based on "full symmetry" which is what the K&W formulation ensures.

4) The "standard" formulation ensures symmetry, making the "simple" solutions and the conditional solutions complementary. Stating the K&W conditions, forcing symmetry, and ensuring the simple and conditional solutions have the same numeric answer is consistent with the bulk of the literature, and seems like a far simpler approach than trying to divide the solutions based on whether the solutions implicitly assume symmetry (which, btw, many conditional solutions do).

5) The quote you refer to was before I'd spent three (!) years writing and rewriting conditional solutions. My current belief is that an approachable conditional solution is in all likelihood easier to comprehend than the "simple" solutions since it does not require the reader to shift from the mental model of picking door 1, seeing the host open door 3, and then deciding whether to switch while standing in front of a closed door 1, closed door 2, and open door 3 showing a goat. I have no empirical evidence of this assertion - but there is plenty of evidence that convincing people the "simple" solutions are correct is extraordinarily difficult. My guess is that these solutions are "simple" only to people who have already "seen the light", and that many proponents of these solutions have forgotten how incredibly counterintuitive these solutions seemed when first encountered.

6) My suggestion is to provide solutions that would appeal to both of the major ways the problem is interpreted. This is both NPOV and likely to be more convincing. I have of late focused my arguments here almost exclusively on NPOV, but I truly think presenting both simple and conditional solutions very early in the article will help rather than hinder most readers. -- Rick Block (talk) 05:01, 25 July 2011 (UTC)

1) I tried to differentiate between saying "indistinguishable" and "undistinguished," the difference being whether it is a distinction that makes a difference. Saying "say" is commonplace mathematical parlance for indicating that it makes no difference; but I acknowledge that the general public ought not be expected to interpret it thusly.

2) I was not proposing to say that such solutions are based on or justified by indistinguishable doors, but that they do not employ distinguished doors. There is a big difference, a difference between original synthesis and objective description. I had not intended to apply this description to borderline cases, but only to clear-cut examples such as the symmetric combinatorial table of vos Savant (who has said, separately, that it makes no difference) and the "combined doors" approach that expressly undistinguishes them by lumping them together.

5) Having spent several years thinking about the problem, you have a very advanced understanding. Reflecting on what you thought of conditional approaches before spending so much time thinking about them may offer a better perspective on making the article readily accessible to readers approaching the subject from a position of ignorance. (More about this in anecdotal remarks below.)

Before I close, I would like to share some anecdotal thoughts about explaining MHP to the uninitiated. Being based on personal experience they have absolutely no relevance for adding information to the article, but I offer them as food for thought anyway.

Having used MHP to entertain and educate several groups of teenagers and young adults, the "explanation" that I have found most effective is isomorphic to the "combined doors" approach. Even adults present are unsure of conditional logic unless they have had some college coursework that covers it. The approach I find most persuasive, after folks have brainstormed a while, is to suggest thinking in terms of "my door" and "Monty's doors." Somehow, the combination of grouping the doors and personalizing them in a possessive sense prompts many people to think carefully about the implications of selective evidence, and this often leads to correct and logically sound answers. Based on this experience, I use MHP as a "teachable moment," not for probability per se, but for thinking about selective evidence in terms of the set selected from and the feature selected for. This does tie to conditional probability, but bringing it up makes people's eyes glaze over. Concluding the session by asking for other examples of selective evidence that can be misleading, and discussing what the evidence really says, can produce entertaining results.

Of course none of this can be used in the article, but as background perspective on making the article understandable it may serve to complement other suppositions that have been offered about how the uninitiated understand explanations of the problem. ~ Ningauble (talk) 18:08, 25 July 2011 (UTC)

Conclusion: Based on the mixed reactions to my original question, I conclude that the answer is no, there is no chance this approach could lead to a way forward. Thank you all for your feedback. I am done here: there is no elephant. ~ Ningauble (talk) 18:11, 25 July 2011 (UTC)

Thank you for your brave attempt anyway. Martin Hogbin (talk) 22:04, 25 July 2011 (UTC)

Probability is completely irrelevant for resolving your dispute, although it is a good thing

You are arguing about the wrong thing. The starting point must be what I know and what do not, and how this affects solution. For suppose I *know* that D1 is winning. Then, obviously, the policy A="choose D1, then stick whichever happens" is the best I can do. But I will not do any worse if I play B="choose D2, then switch whichever happens" (check this!). If I am to some extent unsure that D1 is winning, then B is better than A in any reasonable sense (reasonable means that I want to win). A minor variation is required to discard C="choose D1, switch to D2 if offered, do not switch to D3 if offered". Thus my information about the location of prize (not to say about what the Host is doing) does not matter: notswitching should be discarded. What you are doing on the MHP site is comparable with advising chess-players to decide on the outcome by tossing a coin. What I am saying is a rationale to discard notswitching in a *single* round, if I come to the show once and will never appear there again. With notswitcing discarded, we are left with a choice of a policy out of three. Disputing how to choose one of them is much the same as guessing the roll of a 3-sided die. *Now* thinking of a probability model is a good thing: if some probability model is *assumed* for some reason then *within the frame of the model* to win the prize I should try to minimise the likelihood of the first guess. This can be varied in a usual way regarding known/unknown/known up to a prior, or empirically estimated distribution. This is my POV, you may discard it of course, as switching the POV's is more difficult than switching the doors.RocksAndStones (talk) 15:07, 25 July 2011 (UTC)

Preparing for Wikipedia Content Dispute Resolution

As has been discussed before, I want to take the longstanding difference of opinion about how this article should be written to Wikipedia Content Dispute Resolution and get a ruling so we can inform one side or the other that the consensus of the wider Wikipedia community is against them and that they are not going to get what they want.

For this I need a short description of the differences for the other editors to consider and rule on. So far I have failed to get that, and thus am stalled. So I am changing what I am asking for. I am now asking for any one involved editor to write up a short description of the dispute, and I plan on taking that to Content Dispute Resolution. I no longer care whether anyone else agrees that the description properly describes their position. Everybody had a chance to write up their own short description and failed to do so, so I am moving forward with whatever I have.

I am giving everyone 14 days to write up the short description of the conflict. If I get more than one, I will take a straw poll to see which one I will take to Content Dispute Resolution.

If, after 14 days, I still have nothing, I will follow the example of so many others who have tried to mediate this dispute, call my effort a failure, and quit trying.^{Got the first short description of the conflict, so this option is now precluded.} Guy Macon (talk) 08:54, 23 July 2011 (UTC)

Thank you, Guy Macon. In my view, the controversy is that some insist in presenting only the narrow one-sided view of a solitary source that gloatingly forgot about the famous question "switch or stay", but replaced it with the totally irrelevant question:

After the host has opened "door 3" (not just door Y), what's in maths education the conditional probability of the still closed "door 2" (not just door X)? Cause there could be a slight difference to the "2/3" probability, but within the fixed range of at least "1/2" to "1". And what circumstances could have caused such slight difference?  see The dispute.

This narrow view shows just an irrelevant mathematical side aspect. Repeat: irrelevant, because it never has nor had any influence to the decision that the famous question is still asking for. Readers want to solve the paradox. The fruitless remarkable and controversial history can be shown later in the article. Gerhardvalentin (talk) 15:09, 23 July 2011 (UTC)

I think it would be useful to present the dispute in a way that makes it accessible to as wide a range of user as possible. We do not want to re-run all the same mathematical/philosophical arguments with new people. If we can present the dispute in a way that enables people with no interest or ability on mathematics to contribute it would be an advantage. Martin Hogbin (talk) 16:55, 23 July 2011 (UTC)

Rick's response

The dispute is whether the article should primarily satisfy

1) Wikipedia:Make technical articles understandable), with an initial, extended section focusing exclusively on "simple solutions" that makes no mention of any other solution approaches, in particular the approach using conditional probability. All other approaches will be relegated to later sections of the article intended for experts only. This structural outline (but not the content aspects) are shown in this version of the article.

or

2) Wikipedia:Neutral point of view, with initial sections of the article addressing the most common interpretation of the problem using various approaches specifically including both simple and conditional solutions. The version of the article following the May 2008 FAR (this version) was more or less along these lines, although the "Solution" section in this version of the article arguably expresses a bias in favor of the conditional approach.

I'd be willing to create a version of the article that better exemplifies the #2 approach if anyone thinks this might be useful. -- Rick Block (talk) 17:29, 23 July 2011 (UTC)

That is an attempt to reword the original argument in a way that gives preference to your POV. There is no battle between making technical subjects understandable and NPOV. The simple solutions win on both counts. They are simple, and they are the most notable with the most sources. The rest is a sideshow. Martin Hogbin (talk) 18:12, 23 July 2011 (UTC)

LOL. I assume you're suggesting #1 comes across as biased - however, isn't this exactly what you've been saying for nearly two years (would you like diffs of your own words)? Guy asked for short descriptions of the dispute. If you don't like this one, write your own. -- Rick Block (talk) 19:15, 23 July 2011 (UTC)

I agree that there is no conflict at all (or need be no conflict at all) between Wikipedia:Make technical articles understandable and Wikipedia:Neutral point of view. All editors should concentrate on trying to make the solutions that they understand best or like best as understandable as possible. The perceived conflict between "simple" and "conditional" would then melt away. It's a red herring. The simplists should be aware of the concepts of conditional probability and take care not to write statements which are mathematically speaking false. The conditionalists could think about presenting conditional solutions which build on the simple solutions. Richard Gill (talk) 20:31, 23 July 2011 (UTC)

I worry that, unfortunately, this is preached to deaf ears. The self-evident principle that a simple approach should not be based on – mathematically speaking – false statements, is contradicted here voluntarily and knowingly. Seeing the words simple approach, they insist to put at the same time the words "incorrect argument, that *because* the probability to hit the car in the initial choice of door is 1/3, hence switching gives the car with probability 2/3."  –  That, and only that is what they accept as "the simple approach" or as "the simple solution", and nothing else. As one participant already has repeatedly pointed out in this discussion. See also The dispute.  Gerhardvalentin (talk) 22:20, 23 July 2011 (UTC)

Re: "That is an attempt to reword the original argument in a way that gives preference to your POV", assuming that this is factually correct, I say "Good!" You all had your chance (and still do - for the next two weeks) to create a short description of the content dispute suitable for Content Dispute Resolution that is more to your liking. If the best description I can get is biased, too bad. We can only hope that the outside editors who will be asked to settle the dispute will be able to see past any bias. Or that someone else will write a less-biased description of the content dispute. Guy Macon (talk) 00:23, 24 July 2011 (UTC)

Just to make sure there is no misunderstanding, I am not saying that the description is or is not biased. I am neutral on that question. I am saying that I don't care if it is biased or not. We are going to CDR (Content Dispute Resolution) in two weeks with whatever (as determined by a straw poll) is the best description of the conflict. Note that it doesn't have to be just one; I can present two or even three competing descriptions to CDR if that makes sense. Guy Macon (talk) 00:40, 24 July 2011 (UTC)

I am happy to go along with your approach and I may attempt to to write something that will enable a a wider range of users to contribute. However, I would like to remind you that I originally came to this page as an outside editor to resolve a content dispute. It is quite clear that we will never reach complete agreement unless we all take the view of the original 'page owners'. Martin Hogbin (talk) 09:39, 24 July 2011 (UTC)

As far as I am concerned, your efforts to resolve the content dispute have been very helpful. So have Rick's, Richard's, etc. In my opinion the lack of agreement is not because of any deficiency in anyone's behavior or arguments, but because the content dispute really is is intractable. When editors disagree on content and cannot make progress toward agreement despite good-faith efforts to do so, that's where CDR comes in. Somebody is going to make an argument that the majority of outside editors agree with, and this page will then be edited according to that consensus. Somebody is going to fail to convince with their arguments and that person is simply going to have to accept the fact that they are not going to get what they want. I really don't know which position will prevail - both sides seem to have valid arguments to me. Guy Macon (talk) 17:38, 24 July 2011 (UTC)

Martin's proposal for a content resolution question

Should this article treat the MHP principally as an undergraduate exercise in conditional probability or should it treat it as a simple, well-known, probability puzzle that most people get wrong but which was correctly and simply solved by vos Savant and many other sources and also include a full discussion of all other aspects of the problem for the more specialist reader? Martin Hogbin (talk) 09:35, 25 July 2011 (UTC)

This would be a fine description except for one thing - absolutely no one is arguing that the "article treat the MHP principally as an undergraduate exercise in conditional probability". Other than that one teeny little detail (i.e. that the description of one side of the conflict is an absolute and utter straw man), it's a fine description of the conflict. -- Rick Block (talk) 05:45, 26 July 2011 (UTC)

You insist on giving the undergraduate exercise in conditional probability equal or greater prominence than the either the simple solutions or the wider picture. You want to centre the article around one narrow POV. Martin Hogbin (talk) 08:18, 26 July 2011 (UTC)

Martin. both you and Rick have POVs. As Wikipedia:NPOV tutorial says, "Everybody has a point of view.[...]your view is just one of many possible views that might be reasonably held." I strongly believe that both of you are, in the words of Wikipedia:Neutral point of view, "Editing from a neutral point of view (NPOV), [which] means representing fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources." Your proposed content resolution question is, in my opinion, not suitable. It describes Rick's POV in pejorative terms and described your POV with glowing terms. Please try to describe the content dispute in a fair and unbiased manner. You will have ample opportunity to make your points in your CDR arguments.

Regarding your earlier comment; "I would like to remind you that I originally came to this page as an outside editor to resolve a content dispute." This is true, but your method appears to involve deciding that one side of the content dispute is right, the other side wrong, and adding one move voice to the argument. How has that been working out for you? Has it resolved the content dispute yet? Guy Macon (talk) 10:10, 26 July 2011 (UTC)

All I have done is produce a proposed statement of the dispute for dispute resolution as you requested. Martin Hogbin (talk) 15:07, 26 July 2011 (UTC)

Gerhard's  p.o.v.

"Solving the paradox":  Is it okay that the article should make evident in the first line that the dilemma / paradox is based on the fact that most of us simply overlook and are missing the "underlying development resp. history", and therefore just are judging "50:50" when we see two still closed doors, one of which must hide the only car for sure, and the other one must hide the second goat for sure, period.  –  Forgetting about, that the first door had been selected as "one out of three", and (remember there's only one car)  that the host's two doors must have double chance in this actual game the famous question is about (a game, that - btw - in exactly this manner supposably never was on real stage before nor thereafter    (...as long as not s.o. appears with  "Double chance??? - you could know a little more closely  than just 2/3, if you just knew  what no-one of us knows,  nor will ever be knowing,  but what,  with ease,  could be assumed... ... ...")

Okay that we should forget about useless conflicts, but should favor a multiple (!) approach in "solving"  the paradox?  Including plausible help to understanding by illuminating and not by obfuscating the paradox?

Together with a short example of Bayes' rule in odds-form, showing the conditional probability, just at the beginning? And later on with a link to the article Bayes' theorem? (that's where all of those ineffective mathematical considerations belong, and not elsewhere.)

Based on reliable actual sources, leaving aside unhelpful arguments about that misinterpreted conflict between "conditional" and "simple", and avoiding any misleading statements which are - mathematically speaking - false statements?

These are my questions, and I support to especially refer to the most actual sources, also. Gerhardvalentin (talk) 16:39, 25 July 2011 (UTC)

Once again: We should show a short example of Bayes' rule in odds-form just at the beginning, just as a help to "convince" sceptical readers. But I am clearly and strictly against the article to remain hyped up and applauded for quibbles which had always been and forever will remain completely irrelevant for explaining the famous paradox and completely irrelevant for the decision asked for (switch or stay), messing about with the readers. As serious sources clearly show that the decision asked for never can nor will depend on conditional probability recommending "don't switch" nor on door numbers, and clearly say that conditional probability theory never was nor will be the unique sine-qua-non necessity to find and to give the only correct answer "switch". Reliable sources that give evidence that to switch forever will produce the maximum benefit and to stay never ever can nor will be any indication. We are to refer to reliable sources concerning the famous paradox, not to other kinds of sources that just are using the MHP as a useful example in quite another discipline, missing that, although mathematically correct, all of this is completely irrespective to the famous MHP and its famous question and to the decision asked for. We just should report on that absurd piffling "historical" conflict. Gerhardvalentin (talk) 13:58, 26 July 2011 (UTC)

Richard's p.o.v.

The conflict arises because editors are using wikipedia principles whose interpretation is highly ambiguous in this case. Neutral point of view? Reliable sources? How does one weight academic writers and popular writers? When explaining a popular brain-teaser to children and old age pensioners, is the point of view of writers of university text books on statistics (anxious to sell Bayes' theorem by means of a fun example) an important guide to organisation of the article? Fortunately, while appealing to wikipedia policies has not helped solve disagreements in this case, we are in a situation where, if we wanted to, we could agree to appeal to another authority: the Truth. <shocked silence>. I know that this is strictly speaking not allowed on wikipedia. But we are only talking about elementary logic and elementary mathematics, agreed?. If we could agree to formulations of solutions which are mathematically and logically correct, then we could easily present simple solutions without telling lies to children, and we easily could explain in constructive (layman's) terms what is gained by doing a more sophisticated analysis. We don't have to quote "authorities" when they write sentences which are not strictly speaking true. There are plenty of "authorities" to choose from. Choose formulations which are true, which don't hide things; no need to tell lies for children. The key to this would be realising that Vos Savant asks for an action, not a probability. We only calculate probabilities in order to guide our decision making. It surely cannot be denied that the fact that an "always stayer" wins only 1/3 of the time, while an "always switcher" wins 2/3 of the time is a good reason to go along to the quiz show determined to switch. It also cannot be denied that this conclusion is attained while only making the assumption that our initial door is correct 1/3 of the time, while a conditional probablity solution requires further assumptions, which are certainly not written down explicitly in Vos Savant's question. Oh well, I am surely preaching on deaf ears, again. Richard Gill (talk) 12:44, 26 July 2011 (UTC)

Another willing and thoughtful editor gives up in frustration.

The editor Niguable has already left this discusion page. As have many, many others.

Meanwhile, one editor continues to oppose a consensus that contradicts his page ownership.

Another editor makes rulings that there are no coduct violations taking place. This same editor rules that a third editor is not as pious as he, due solely to the length of time of that editor's involvement in these discussions.

Perhaps the article has gained a second Page Ownership violating editor?

75.33.51.159 (talk) 14:34, 26 July 2011 (UTC)

Please do not attack other editors. Comment on content, not on contributors. Personal attacks damage the community and deter users. Please stay cool and keep this in mind while editing. Thank you. --Guy Macon (talk) 08:06, 27 July 2011 (UTC)

To be clear, in writing "I am done here" in a thread above I meant that I was done there, done with the question raised in that thread. I have been here, following this article and occasionally participating, for more than three and a half years. If I have chosen to speak up only on rare occasions when I thought I had something constructive to offer, something potentially more meaty than chewing old bones of contention, it does not mean I am not here, just that I don't have such brainstorms very often. ~ Ningauble (talk) 12:02, 28 July 2011 (UTC)

Perhaps I have misinterpreted the meaning of "There is no elephant." 76.190.251.93 (talk) 13:19, 28 July 2011 (UTC)

Other corrections needed: here is zero evidence of any consensus. Nobody here has posted any "rulings," only observations and opinions. [Comment Deleted] Guy Macon (talk) 19:17, 28 July 2011 (UTC)

Guymacon, I think you challenged me when I made a comment similar to this. If you suspect sock puppetry go through the proper channels. Martin Hogbin (talk) 22:36, 28 July 2011 (UTC)

You are entirely correct, and I do appreciate the correction. Guy Macon (talk) 02:15, 29 July 2011 (UTC)

Ningauble, your views on this subject seem not too dissimilar to mine. Are you willing to discuss this, either here or elsewhere? Martin Hogbin (talk) 22:38, 28 July 2011 (UTC)

"Discuss this" is too nonspecific for me to formulate a response. Regarding the subject of this thread, which is nominally a meta-discussion of my participation here, I have said about as much as I am able to articulate. Regarding discussion venues I think centralized discussion is better. Although I have responded to enquiries on my talk page, I would prefer it not become a fork of this discussion page. ~ Ningauble (talk) 16:01, 29 July 2011 (UTC)

Veritas

Regarding the deletion of this section, the original reverter did not discuss his reasons. Invoking "personal attacks" is a unique interpretation of an edit that doesn't name a single editor. It's equally or more likely that the reverting editor doesn't care to acknowledge the accuracy of the post, and used a contrived excuse to revert it. 166.216.194.35 (talk) 05:58, 27 July 2011 (UTC)

Please remember to assume good faith when dealing with other editors. Thank you. --Guy Macon (talk) 08:06, 27 July 2011 (UTC)

Two ways to look at the MHP

Ninguable (and anybody else) I believe there are two ways to look at the MHP. The first is as a simple probability puzzle. In this case there is a long standing convention to make the necessary assumptions to keep the problem simple (for example that the host chooses evenly when he has a choice and always offers the swap). This is clearly how the problem was intended both by Whitaker and Selvin, as a simple brain teaser. Whitaker did not mention door numbers in his letter to vos Savant so we assume that he did not think the individual doors chosen to be important. Vos Savant unfortunately added the door numbers, intending to clarify the problem. She later recognised this as a mistake. If the problem is taken as a simple brain teaser, the simple solutions are fine.

The other way to look at it is as a serious question. Suppose you were a probability consultant and a client asked you Whitakers question. The solution is now much more complex. You would have to start by asking your client a whole bunch of questions to find out exactly what they wanted to know. These would include things such as, 'Do you want an answer from the perspective of a player on the show?', 'Do you consider the door numbers to be significant?', 'Do you want to know the best strategy to win?', and many more.

Do you agree that these are the two ways to look at the problem? Martin Hogbin (talk) 23:56, 29 July 2011 (UTC)

No, Martin, I disagree. There are as many faces as one can imagine. There is a situation which can :be modelled and discussed, under assumptions required for particular solution. I am very :surprised that nobody in this dispute came up with the solution like "If you know where is the :prize just pick the right door". This problem will *never* be done, and new people will come with :a frish look, very different from the persisting stagnation promoted by the majority of the :editors of this page. The dispute about "correct formulation" of a losely posed problem is a :ridiculous scholastics. In my POV, you disregard my words and that of some editors (notably :Richard Gill) willing here to promote clear views and structure comparable with any serious :mathematical article. The paradox itself that "it is not 50:50" is completely resolved by vos :Savant's argument irregardless of her further opinions about the door labels. If you as most of :practitioners of the MHP promote the probabilistics views, then do not forget to explain to a :laymen what *is* the probability, which is by no means primitive, especially if you appeal to :conditional probs. I had an option to discuss the issue with algebraists, they were slow to grasp :the things this way. The combinatorial viewpoint, which was explained in this discussion several :times is left by you without comment and attention. What does it mean? You did not take care to :look in the argument, or is it so exciting that it is best to wait and see how :other people will :react? You know that the host (as door opener) is a dummy player, he can neither

help by signals nor cheat you by clever door-opening. His behaviour is irrelevant, and this :implies everything you wish to achieve with conditional probs. Please respond if you see the point.

And if you are interested to see new faces of the problem: nothing can be easier. Just open Olle :Haggstrom's textbook Streifzuege..., look at the game matrix, and explain how the structure of the :polytope spanned on the rows (columns) reflects in the 2/3 game value. This is just one of the :faces a quality mathematical article is expected to have. My largest disagreement, however, is the :modus operandi. Instead of having 5-10 drafts to put them together this discussion continues :(very interesting and exciting) farce.RocksAndStones (talk) 06:44, 30 July 2011 (UTC)

Yes, of course there are very many faces to the MHP. Our job is to explain all this to an audience of widely varying interests and abilities. There is an unwritten assumption for mathematical puzzles that you take the problem in the simple way that it was intended. Sometimes, such as with the Two envelopes problem this step is not so easy because the informal language used to describe the puzzle is not capable of defining a precise problem. However, in the case of the MHP there is a simple interpretation of the problem that was undoubtedly the one intended by Selvin, that we know was the one originally intended by Whitaker, and was formulation addressed by vos Savant in her answer. That is where we should start in our article. It is the simplest and most common formulation with the simplest and most common solution.

After the puzzle is out of the way, I completely agree with you. You first have to ask what is even meant by 'probability'. Are we talking about subjective/Bayesian probability where our answer is based on the the information know by a particular individual (such as the player), or are we talking about an objective/frequentist meaning of the word where we envisage a repetion of some process and we consider the relative frequency of certain outcomes? These interpretations may, in the end, result in the same answers but serious discussion is not possible without agreement as to the exact subject of interest.

My point is that a few long-term editors insist that one specific interpretation of the problem and one specific solution to that problem is 'the right one' and that this fact is so important that we can't even talk about the simple solutions without mentioning it in some way. As an answer to a simple mathematical puzzle the simple solution is just fine; as a realistic solution to a real world problem the simple solutions are deficient in dozens of ways. Why give just one of these ways undue prominence? We should discus all these issues in a scholarly manner, after we have answered the simple puzzle Martin Hogbin (talk) 10:16, 30 July 2011 (UTC)

Martin, OK though no answer to my direct quest. Let me explain to be on sure side:

under no circumstances is there a way to win (sometimes, perhaps depending on the history) three :::*cases* out of three. And you have a strategy to make 2 out of three. This is the whole puzzle,

both conditional and unconditional, if you prefer these terms. Everybody willing to attribute :::probs to the cases can do it -- this is secondary, and probs need not be equal. Moreover, :::instead probs you can attribute money value to the cases: all what is needed is additivity.

Of course, the "simple solution" (in my terms : you can make 2 out of three) is the one which :::must stay first, to explain that 50:50 is illusory. This is what everybody understands.

Then we should proceed to discussing why the constant-action always-switching policy is optimal, :::and in which senses it is optimal.

Thus, I see a consensus: you, me, Richard and perhaps Ningauble. Apparently Rick Block and many :::others *love* struggling and will struggle till the end of the world, resolving ambitions on the

level of secondary school math. If we can agree now to draft a reasonable write-up: let us do :::it. If the remaining editors do not want to cooperate -- we can arrange article 3-Door-problem :::in a way we find optimal, leaving some others to computerise their comments to instruct each new :::naive visitor why one needs to multiply 1/3 and 1/2.RocksAndStones (talk) 12:39, 30 July 2011 (UTC)

I agree with you that, 'Of course, the "simple solution" (in my terms : you can make 2 out of three) is the one which :::must stay first, to explain that 50:50 is illusory'. I also suggest that it is a very bad idea for individual editors to complicate this solution with their own pet formulations of the problem or personal interests. The simple solutions stand alone as the answer to a simple puzzle.

After that, there is plenty to discuss. Martin Hogbin (talk) 13:52, 30 July 2011 (UTC)

Citation: "a few long-term editors insist that one specific interpretation of the problem and one specific solution to that problem is 'the right one' and that this fact is so important that we can't even talk about the simple solutions without mentioning it in some way". If you mean conditionalist's approach (that one must for some reason address the problem of odds in the famous situation), then this approach will be gradually pushed in the corner, as it is only suitable for undegraduate probability texts. A layman (in wide and positive sense of the word) cannot understand that the odds depend on the behaviour in the situation when there is a freedom of choice of the door-opener, thus for the general public this approach is a dead-end. Moreover, the conditionalism adds nothing to the dilemma, as the assumption of coin-tossing host is superfluous. Have you constructive suggestions how to proceed, as it is impossible to convince somebody not willing to get convinced.RocksAndStones (talk) 15:59, 30 July 2011 (UTC)

Martin has many times posted a fairly detailed proposal of how the article should be structured. Rick is strongly opposed. I think a majority of presently active editors are for Martin's proposal, if only so that we can say goodbye to the conflict and get to work. That's why I proposed that Martin's proposal be the topic of official "content resolution". Guy seemed to ignore my proposal, I don't know why. He wants names for the two positions, but why not just call them Martinist and Rickist? There will be a decision made by a bunch of wikipedia editors without any particular knowledge or even interest in this particular problem, but who have some kind of authority based on long time work for wikipedia which has been valued by the community. One or more people will be dissappointed, one or more will be pleased, but the main thing is that the thing is settled for a while. Richard Gill (talk) 07:35, 31 July 2011 (UTC)

Martin's plan is incomplete, but it certainly can be taken as a base. Dividing the editors in parties, as suggested above is not a good idea. I am convinced that when the process will start moving the opposition (if any) will dissolve in the air, or will take the normal working attitude. In particular, Rick with his 4 yrs experience in answering amazing laymen questions will certainly be pleased to provide invaluable help, M I right, Rick? As for Guy Macon, his love to order (exhibited in correcting punctuation and indentation of some sloppy mathematicians) might be a huge support too. Now the question is how to settle this technically. We need a draft and (old and new) illustrations. Unfortunately, I am a LaTex man so somebody, not me, should take care of the pictures. Guy Macon, could you help?RocksAndStones (talk) 10:40, 31 July 2011 (UTC)

I'm willing to provide whatever help I can - however I strongly disagree with Martin's outline for reasons I've stated numerous times (specifically that it violates WP:NPOV). As I read the sources, there is a bright line between those that answer the question of whether a preselected strategy of switching is better than a preselected strategy of staying, and those that answer the question of whether a player having initially chosen Door 1 and then having seen the host open Door 3 should switch to Door 2. Starting the article with an extended section based on those sources espousing the former view without any mention whatsoever that other views even exist (because we assume our readers are too stupid to understand the difference) strikes me as endorsing one view at the expense of the other and amounts to a willful violation of one of the fundamental principles upon which all Wikipedia content must be based. Not only are both of these views well represented by highly numerous reliable sources (making this an NPOV issue), but nearly all people initially reading the problem (97% per Krauss and Wang) clearly focus on the latter view. Of course, you can attempt to change the reader's focus from the specific case to the consequences of a preselected strategy and once accomplished this makes the problem "simple" - but effecting this change of focus (particularly without directly discussing it) is far from easy.

As I've also said numerous times, it's not only obviously more in keeping with NPOV but I think more convincing as well to present solutions addressing both questions early in the article (without insisting that one or the other view is "more correct"). Martin's approach asserts the "simple" view is most correct (as it is presented first and without any qualification). The text in the solution section as of the 2008 FAR version of the article arguably says the conditional approach is more correct. We could eliminate any hint of bias with a completely neutral transition between these approaches, i.e. something like "Another approach is to determine the conditional probability of winning by switching given the player has initially selected Door 1 and the host has opened Door 3. Referring to the figure above ...". -- Rick Block (talk) 17:48, 31 July 2011 (UTC)

Rick, you say "As I read the sources, there is a bright line between those that answer the question of whether a preselected strategy of switching is better than a preselected strategy of staying, and those that answer the question of whether a player having initially chosen Door 1 and then having seen the host open Door 3 should switch to Door 2." This bright line which you see is a shining bright red herring. You see it, a few authors see it, but a lot of people and a lot of writers don't. And it is not what we are talking about! This is not the important distinction! There are other preselected strategies than the two you mention: "choose Door 1 and switch, whatever the host does" and "choose Door 1 and stay, whatever the host does". The strategy "choose Door 1, watch which door is opened by the host, and only then decide to stay or switch according to the conditional probability that the car is behind the other door" is also a preselected strategy: we can imagine either door being opened by the host and we can imagine both computations and both conclusions, in advance. The strategy "choose door 1 and see which door is opened, then toss a coin whether to stay or switch" is also a preselected strategy. When we talk about strategies we do not restrict ourselves to the two rather special and extreme strategies which you think is the subject of the simple solutions. We also include your favourite stategy, and we also include the obvious strategy following the "it doesn't matter" answer to the question whether or not you should switch.

Now, it is child's play to see that any strategy which would in some circumstance lead you to stay is beaten case by case (ie where-ever the car is, and what-ever the host does) by an appropriately coupled strategy of always switching. So one can *in advance* decide on totally rational grounds to only consider the three strategies: choose Door 1 and always switch; choose Door 2 and always switch; choose Door 3 and always switch (and also, randomized choices from these three strategies). You, Rick, forget that the *only* reason for determining your action via conditional probability is because this is a way which gives a guarantee that your strategy can't be improved; ie, its overall, *unconditional*, win chance cannot be improved. My apologies that the writers of introductory text books on probability theory don't often mention this explicitly. Probabilists in general know this so well that they don't bother to explain it to other folk. Please, please, realise that checking conditional probabilities is not necessarily the only way to get this guarantee! Any way to show that your overal win-chance can't be improved above 2/3 is sufficient to prove the optimality of "Choose Door 1 and switch whatever". A little thought in advance shows us than in the case of MHP, we may completely forget about staying, in any circumstances. However the car is hidden, whatever the host does. The only thing one should pay some attention to is, which door to choose at the start. If all doors are initially equally likely to hide the car, then the three always switch strategies and all randomized combinations of them all have overall win chance 2/3. Since there is no point whatsoever in considering any other strategies, this proves that "choose a door and switch whatever" is the best you can do. As a corollary, it follows that in this case all conditional probabilities must support switching. There is no need whatsoever for Bayes! No need whatsoever to compute them! No need whatsover to worry about possible host-bias! A little strategic insight is enough.

Sure, wikipedia has to follow the reliable sources and reliable sources are typically ten years out of date. But remember that reliable sources have their "sell-by" date. The conditional probability approach need only be a foot-note for specialists. It's foolish to highlight it in the article. Richard Gill (talk) 18:56, 31 July 2011 (UTC)

I want to put the simple approach first not because it is 'correct' but because it is simple. That is how most technical subjects are treated. Once you move away from the simple mathematical puzzle aspect of this problem there are many problems, questions, formulations and solutions that arise. Your preferred approach is just one way to tackle the problem. Why should we single your preferred approach out for special treatment? If we are going to say, '"Another approach is to determine the conditional probability of winning by switching given the player has initially selected Door 1 and the host has opened Door 3' at the start of the puzzle, why do we not also say 'and another approach is to use game theory... and another approach is to consider the symmetry of the situation... and another approach is to consider the Bayesian perspective of the player .... and another approach... '. That would be absurd. Let us get the puzzle bit out of the way then have a proper scholarly discussion of the wider aspects of the problem. Just to have one special case is just your POV. Martin Hogbin (talk) 19:00, 31 July 2011 (UTC)

The "simple" approach is arguably no more accessible than the conditional approach (the decrease in complexity is offset by the need to change the mental model of the problem). I have argued for including the conditional approach not because it is my preferred approach, but because it is extremely prevalent in the literature (much more so than game theory, Richard's "switching beats any other strategy" approach, etc.). WP:NPOV demands that the article "fairly represents all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint". Based on my reading of many, many of the sources, my opinion (not my POV) is that the conditional approach is at least as prominent among reliable sources as the simple approach. Putting the conditional solution on a par with the simple solutions is simply fairly representing its prevalence. -- Rick Block (talk) 02:26, 1 August 2011 (UTC)

No approach comes close in number of sources to the simple solutions. Martin Hogbin (talk) 09:04, 1 August 2011 (UTC)

Rather than continue this argument (again), I'll simply refer the interested reader to a previous time we've discussed appropriate and inappropriate ways to weigh the prevalence of sources, see [3]. -- Rick Block (talk) 15:15, 2 August 2011 (UTC)