User talk:Nijdam/Discussion

Discussion on the Monty Hall problem.

The MHP

1. A player stands in front of three closed doors. behind one of them is a car, behind the others goats.
2. The player may choose a door and have what's behind it.
3. From this setting ordinary people conclude that the car is placed randomly. Call C the door with the car: P(C=c)=1/3.
4. And furthermore that the player is completely unaware of the position of the car. Call X the chosen door: C and X are statistically independent (of each other).
5. Suppose the player chooses door x; as typical example take x=1.
6. What is the probability the chosen door hides the car?
7. This is a tricky question: many people will think it is the probability the car is behind door 1: P(C=1)=1/3.
8. The actual meaning must be:P(C=1|X=1), accounting for the event X=1.
9. Due to the independency: P(C=1|X=1)=P(C=1)=1/3.
10. Before the chosen door is opened, the host opens one of the two other doors, and exhibits a goat. Call H the opened door.
11. Strategy of the host: always opens one of the remaining doors, always showing a goat.P(H=x}X=x)=0; P(H=c|C=c)=0.
12. As an example take H=3.
13. The player is offered to switch to the other closed door.
14. The player now sees only two closed doors, between she has to choose.
15. At this point many people think the odds are equal.
16. The player has to base her decision on the probability in this situation the car is behind door 1 (or door 2).
17. This probability is: P(C=1|X=1,H=3)
18. To find the numerical value of this probability we need info about what the host will do when he has to choose between two doors with a goat.
19. It seems likely the host then chooses randomly: P(H=2|X=1,C=1)=P(H=3|X=1,C=1), etc.
20. Then with Bayes' formula or using arguments of symmetry, it follows: P(C=1|X=1,H=3)=1/3.
21. This is the same value as the unconditional probability.

The point of the MHP

• Almost no one doubt the fact that a door is actually seen opened. It is the charm of the puzzle.

• The paradox of the MHP is that a lot of people think it is irrelevant whether to switch or not, as they think the odds are equal. This also implies the problem is considered to be a probabilty puzzle.

What to base the decision on

The decision asked for is whether to switch to door 2 or to stay by door 1, after door 1 has been chosen initially and door 3 has been opened showing a goat. Switching will be the right decision when door 2 has a greater chance on the car than door 1. Under the normally accepted assumptions, the original chances (or the chances after picking door 1) are 1/3 - 1/3 - 1/3 for all doors. The opening of door 3 provides information. The (new) chances are 1/3 - 2/3 - 0 for the 3 doors. Hence in the new situation it is better to switch.

Combined doors

The reasoning is as follows: The chosen door has probability 1/3 to hide the car, hence the two other doors have probability 2/3 to hide the car. When one of these doors is opened, this door has probability 0 to hide the car. Hence the other one must have probability 2/3 to hide the car.

The error lies in confusing the probabilities before opening a door and there after. The fact that for the opened door the "probability" of hiding the car is 0, means the considered "probability" is the conditional probability given the opened door. To decide that the remaining door has "probability" 2/3, we need to calculate the conditional probability. Unconditional the two not chosen doors have together probability 2/3. But it is not immediately clear that conditional this sum is also 2/3. This has to be proven.

Another way of understanding the flaw is: choosing for both the initially not chosen doors is not the same as switching to the remaining unopened one.

I asked you if you see the difference between the prior and the posterior probability. You argued about how the posterior probability is calculated. And I explained it's unimportant. I also tried to make clear to you that two probabilities may have the same value and yet be different!

I also explained to you that conditioning means an essential reduction of the sample space. After some discussion I replied: It seems you do not understand what is meant by the term condition. You have your own ideas about conditioning, but they do not coincide with the probabilistic view. So it is useless to discuss this point, where your and my definition differ. Anyway, my definition is the probabilistic one. Maybe we better speak of probability before and after an event has happened.

Simple solution

Simple solution (A): the chosen door has, due to the random placement, 1/3 chance on the car, the opened door has chance 0, hence the remaining door must have chance 2/3.

Simple solution (B): the chosen door has, due to the random placement, 1/3 chance on the car, hence switching will get the car with chance 2/3.

(A) is incorrect, because a logical error. It may however be interpreted different and as equivalent to (B).

(B) is a correct way of reasoning, but not a solution to the (usual interpretation of the) MHP. It solves a (slightly) different problem.

Let us assume the player has originally picked door 1, and we relate all to this situation.

As soon as one speaks of the opened door having 0 chance on the car, this means the conditional probability.

Somewhere in the discussion we saw that P(C=1)=P(C=2)=P(C=3)=1/3 and P(C=3|H=3)=0, showing the change in nature of the concerned probabilities after the opening of door 3.

Also: P(C=1|H=3)+P(C=2|H=3)+P(C=3|H=3)=1, and because P(C=3|H=3)=0, it follows that P(C=1|H=3)+P(C=2|H=3)=1. And although we know P(C=1)=1/3, we don't know at forehand the value of P(C=1|H=3). It is however easily calculated in one or another way.

From this you see that the simple solution is not correct in the (normal) situation where you see the doors pointed at and opened.Nijdam (talk) 20:54, 31 May 2010 (UTC)

I am not sure why you say that the simple solution is not correct. We can say in words that the probability that the car is behind door 1 is unchanged when door 3 is opened to reveal a goat because we know by symmetry that the probability the car is behind door 1 is the same whether door 2 or door 3 is opened and these two probabilities must average out to the original probability. This is the same as the above but in words.

The probability is not unchanged! as we discussed, it has changed its nature and has the same value as before the change.

NB Suppose your participation in a bunch of shares is 1/3. Now I change something in the bunch, but keep your participation at 1/3. Would you agree with my change, because you argue that for you nothing has changed?

Remaining door

Another erroneous way of looking at the MHP is introducing the variable R = the remaining closed door (after a door has been chosen and one opened by the host). With C=door of car, X=chosen door and H=door opened by host, one may argue that X, H and R are all different and because P(C=X)=1/3 and P(H=C)=0, P(R=C)=2/3, hence the simple solution should be correct. The error lies in the use of P(R=C) to base the decision on, instead of P(R=C|X=1,H=3).

Writing the "remaining door" is not unequivocal. It may, as per Richard's definition, refer to a random variable. It may also refer to door No. 2 in the usual version of the MHP.

Whitaker

Okay, let's start with Whitaker.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? (Whitaker 1990)

Pretty clear: suppose you're the player. You are on stage, pointing to door No. 1, and the host has just opened door No. 3, showing a goat. You are offered the choice of either what's behind door No. 1 or door No. 3. Implicitly it is supposed the car is placed randomly and your choice and the position of the car are independent. There is also a strong suggestion that the host deliberately showed a goat, and if needed randomly. It is now up to you to calculate your chances. Nijdam (talk) 20:42, 31 May 2010 (UTC)

Are you saying that if someone asked you this question, in these words, that is what you would assume they wanted to know?

Of course.

You are kidding me? Are you telling me that if say, an industrial client, asked you a rather ambiguously phrased question you would just give then the answer to the exact question that they had asked rather than try to determine what it was that they actually wanted to know.

Using my example, how would you answer the question, 'What is the probability that a person is male?'.

Why would somebody want to know the result if they had chosen door 1 and seen door 3 opened?

That's the question.

It is indeed the question and it has no reasonable explanation. What possible reason could there be for a person writing to a magazine to ask a question about 2 specific doors? far more likely he wanted general advice.

Might in not be possible, I would say likely, that they actually just wanted to know if it is better, in general, to swap or not? Martin Hogbin (talk) 21:13, 31 May 2010 (UTC)

No, not in general. It is likely he wanted to know whether to swap in every possible situation, with a chosen door and an opened one. And the presented situation stands as an example. Nijdam (talk) 10:41, 1 June 2010 (UTC)

So you are saying that Whitaker actually meant to ask the question 'I choose one specific door and the host opens another specific door to always reveal a goat' rather than 'I open any door and the host opens any of the remaining two doors to always reveal a goat'? Martin Hogbin (talk) 10:47, 1 June 2010 (UTC)

No, if you find yourself really in the described situation, you actually point to a door and see one opened. And also is it not clear on forehand that every combination of doors leads to the same conclusion. This needs some reasoning using symmetry arguments.Nijdam (talk) 21:35, 31 May 2010 (UTC)

Whitaker did not find himself in that situation. He is asking a hypothetical question about what a player do in some imagined future situation. He is not a statistician, just a member of the public. Martin Hogbin (talk) 23:01, 31 May 2010 (UTC)

Let me ask you: Is it to your advantage to switch your choice? Answer yes or no.Nijdam (talk) 10:36, 1 June 2010 (UTC)

In what exact circumstance? Martin Hogbin (talk) 10:47, 1 June 2010 (UTC)

Your answer to Whitaker's question. Nijdam (talk) 11:21, 1 June 2010 (UTC)

You are asking me how I would answer Whitaker's question, OK.

Firstly, if there was any opportunity to ask further questions to clarify what exactly he wanted to know, I would consider it absolutely essential to ask those questions before giving an answer. No doubt you want me to assume that no such opportunity exits.

The next question you need to answer is through what medium am I to give my answer. If it was through the medium of a column in a popular general interest magazine, I would give an answer along the lines of vS.

So my answer would be along the lines of:

State my assumptions: host always opens an unchosen door to reveal a goat and always offers the swap. The car is initially placed uniformly at random and the host chooses a goat door uniformly at random. I would the point out that the last assumption had the effect of making it unimportant whether the player had to choose before or after the host had opened a door. (If only vS had done this, this whole argument would not exist.)

Finally I would give a simple solution concluding that the player has a 2/3 chance of winning by swapping.

If, however, there was room for a longer answer, say as an article in a popular science magazine I might give a more detailed discussion of the subject. Martin Hogbin (talk) 10:34, 7 June 2010 (UTC)

Sorry Martin, I just wanted to hear yes or no. Nijdam (talk) 11:16, 7 June 2010 (UTC)

I cannot answer that question without some assumptions. Assuming the host always opens an unchosen door to reveal a goat and always offers the swap, yes. I guess that I should add, assuming the car is placed uniformly at random and that, if you want to be fussy, assuming that the host does not always choose door 3 when possible under the above assumptions. Martin Hogbin (talk) 11:46, 7 June 2010 (UTC)

I made already the necessary assumptions. So your answer is yes. Alas, this is the wrong answer, as the car is behind door No. 1 (being the host, I know this).Nijdam (talk) 16:16, 7 June 2010 (UTC)

You seem to be just telling me that probability is a state of knowledge, something I know perfectly well already. Based on my stated assumptions my answer is correct. This is the very point that I made long ago. Martin Hogbin (talk) 16:24, 7 June 2010 (UTC)

Am I guessing right that, had you know the car is behind door No. 1, your answer would have been no?Nijdam (talk) 16:43, 7 June 2010 (UTC)

Correct, It is not to my advantage to switch my choice if the car is behind 1 and I wish to have the car, Martin Hogbin (talk) 16:54, 7 June 2010 (UTC)

So may I say, you take notice of the information available to you; as you said: your state of knowledge?Nijdam (talk) 23:08, 7 June 2010 (UTC)

Yes, of course. Martin Hogbin (talk) 23:31, 7 June 2010 (UTC)

Well ... then why won't you take notice of the knowledge of the opened door? Nijdam (talk) 10:03, 8 June 2010 (UTC)

What does that tell me as a contestant? Clearly it affects the probability that the car is behind the opened door and the probability it is behind the unopened, unchosen door, but it tells me nothing about the probability the car is behind the originally chosen door because I have no idea of the host's possible goat door preference, I can only apply the principle of indifference to this choice and take it to be uniform. To put it another way, the player's state of knowledge about the probability that the car is behind the originally chosen door does not change. Sure, the player's total state of knowledge does change but that is not important. Martin Hogbin (talk) 15:28, 8 June 2010 (UTC)

As you say, it clearly affects the probability for the opened door. But why does it affect the probability for the unopend one? Nijdam (talk) 07:05, 9 June 2010 (UTC)

That is the MHP. We know from various arguments, some involving conditional probability, that, after the host has opened door 3 to reveal a goat, the probability that the car is behind door 1 is unchanged but the probability it is behind door 2 is now 2/3, the question is how best to show this.

One simple argument to say that the opening of door 3 does not affect the probability that the car is behind door 1. We know the car is still there, so the total probability that the car is behind any door must be 1. Opening door 3 shows us that the probability the car is behind door 3 is 0, this it must be behind door 2 with probability 1 - 1/3 = 2/3.

The point I would like to make here is that any method of solving the problem must involve some logical reasoning. In your preferred method of solving the problem you have to show that all the elements of your sample set are independent. How do you do that? Only by logical reasoning. Martin Hogbin (talk) 11:04, 9 June 2010 (UTC)

I only needed to know from you: We know from various arguments, ... etc. As you say, we need arguments! It is not sufficient to say: the car is with probability 1/3 behind door No. 1, hence it is with 2/3 behind the other unopened door. BTW: I don't understand your second section. Please read it yourself and see if something is missing there. About the rest later on. Nijdam (talk) 12:55, 9 June 2010 (UTC)

Maybe some formalism might help: call the probabilities of the car behind the different doors: b1, b2 and b3. Then b1=b2=b3=1/3. Now where do I have to fit in the zero probability of the car behind door 3 after this door has been opened? Clearly it differs from b3! And the same for door 2. It is a new situation with new probabilities. Call them a2 and a3; a3=0 and a2 is the one to be calculated. How? Please, your turn. Show me the calculation of a2. Nijdam (talk) 08:02, 10 June 2010 (UTC)

a1 + a2 + a3 = 1 (The car is still known to be present) a1 = 1/3 (shown previously) , a3 = 0 (a goat is revealed behind door 2), therefore a2 = 2/3 Martin Hogbin (talk) 09:30, 10 June 2010 (UTC)

Shown previously? Please show me again that a1=1/3, because that's what it is all about in our discussion.Nijdam (talk) 09:48, 10 June 2010 (UTC)

We know from symmetry that P(C=1|H=3) = (C=1|H=2), we also know that P(C=1) = 1/3 and that ( P(C=1|H=3) + P(C=1|H=2) } / 2 = P(C=1) from symmetry and that fact that H=2 or H=3 are the only two options. Thus P(C=1|H=3) = a = 1/3. Martin Hogbin (talk) 10:32, 11 June 2010 (UTC)

Yes, yes, but where does the simple solution fit in?Nijdam (talk) 17:00, 11 June 2010 (UTC)

I have shown these calculations in a standard mathematical format to make clear to you what I am talking about but a person whi is not familiar with conditional probability might use the same reasoning informally. For example they might reason that it is obviously unimportant (by symmetry) which door the host opens thus a diagram that does not specifically refer to any door is correct. Now you might argue that they are in fact using conditional probability unwittingly but the simple solutions shown will satisfy most people and are in fact correct. We are letting people take a mental short cut, knowing that it is justified. Martin Hogbin (talk) 18:07, 11 June 2010 (UTC)

To start with: do you agree the simple solution does not provide the calculation of a1? Nijdam (talk) 20:53, 11 June 2010 (UTC)

No calculation is required. The only new information revealed when the host opens a door is the number of the door that he chooses. If this is done uniformly at random then it is obviously unimportant which door it turns out to be. So a1 = b1 = 1/3. Martin Hogbin (talk) 22:24, 11 June 2010 (UTC)

I deduce from your answer that you agree that no calculation is provided. Do you agree that b1 and a1 are different (things)?Nijdam (talk) 23:10, 11 June 2010 (UTC)

(editing)

'Different things' has no well defined meaning here. You know full well that I understand the difference between prior and posterior probabilities. Your a1 and b1 are such and are quantities which potentially might have different values. However it is quite obvious (and proves to be correct in a more formal analysis such as you would like) that they have identical values and can be used interchangeably in the symmetrical case. Let me repeat that the use of symmetry is a common, powerful, and respectable, method to simplify mathematical or logical problems and avoid the need for complex formal analysis. Martin Hogbin (talk) 10:41, 12 June 2010 (UTC)

Although you hesitate to admit they are 'different things', you do admit they are 'not the same thing'! And, yes, they have the same value. I have two sons and two cars, yet I like to think that the number of sons I have, is a different thing than the number of my cars. On some tax form, inquiring after the number of children I have, I'm not allowed to respond by writing "I have two cars". They are not interchangeable. Neither are a1 and b1. Symmetry doesn't help here. Symmetry comes in to ease the calculation of a1.

Would you say that the simple solutions is correct and complete for this case.Usual rules, car placement and goat door choice random. The player chooses door 1 and the host opens either door 2 or door 3 to reveal a goat but neither we nor the player know which door. Martin Hogbin (talk) 17:25, 12 June 2010 (UTC)

I take it that you mean the situation as I formulated it on the MHP discussion page: the host reveals his intentions, and before he actually opens a door the player has to decide. Yes, then the simple solution is correct.Nijdam (talk) 09:12, 13 June 2010 (UTC)

I was intending to go one step further. The host has actually opened a door but no one else knows which one. Is the simple solution correct now? Martin Hogbin (talk) 09:23, 13 June 2010 (UTC)

That is the problem with interpreting probability. Or better, what you will consider to be probability. For the player the situations are equivalent. In repeating, the host may (has to) open now and then door No. 2. So he might as well just announce his intentions. The probability does not refer to the actual situation (as I showed you somewhere above, with the car behind door No. 1) but to the circumstances. Nijdam (talk) 09:47, 13 June 2010 (UTC)

Yes, I already knew what you kindly told me, but what is the answer to my question?

I did answer your question: the simple solution solves this case. I only elaborated a little to make sure you understand that repeating of the experiment means the host opens a door according to his strategy and we don't know which one. Nijdam (talk) 21:23, 13 June 2010 (UTC)

What argument would you use then to show that the probability that the car is behind door 1 remains unchanged after the host has opened a door? Martin Hogbin (talk) 10:15, 14 June 2010 (UTC)

It is not the argument we may use, but the different type of probability what it is about, remember? You said yourself: prior to the opening: b1=1/3; posterior: a1=after calculation or argumenting=1/3. Nijdam (talk) 15:19, 14 June 2010 (UTC)

Really Martin, answer this question: what upon does the player base her decision: the probability b1 or a1?Nijdam (talk) 15:34, 14 June 2010 (UTC)

I am not sure what you are getting at here Nijdam. The player must, of course, always decide in the latest available probability, which you called a1. The probability that the car is behind door 1 after the host has opened either door 2 or door 3 in this case. As you said the simple solution was correct in the above case, I presume that you accept that the the probability of finding the car behind door 1 after a door has been opened (a1) is the same as that before (b1). Martin Hogbin (talk) 17:58, 14 June 2010 (UTC)

Before I answer your question, what do you mean by: The probability that the car is behind door 1 after the host has opened either door 2 or door 3 in this case. It is easily written down in words, but I like to see it in a formula, to be sure of its meaning. Use the variables X, C and H.Nijdam (talk) 21:06, 14 June 2010 (UTC)

I mean do you agree that P(C=1| H=2 OR H=3) = P(C=1) = 1/3 Martin Hogbin (talk) 21:31, 14 June 2010 (UTC)

More precisely, given {X=1}: P(C=1|H=2 OR H=3})=P(C=1), formally: P(C=1|X=1, H=2 OR H=3})=P(C=1|X=1)=P(C=1). Okay. Nijdam (talk) 09:35, 15 June 2010 (UTC)

Now you define given {X=1}: a1=P(C=1|H=2 OR H=3}), but I, as you may read above, define, given {X=1} a1=P(C=1|H=3}). They're different. It comes down of course to the question whether the player actully sees a door opened, or just knows of the intention of the host to open a door. It is about time you accept that almost no one doubt the fact that a door is actually seen opened. It is the charm of the puzzle, it is complete obvious that the host opens the door before he makes the offer. Why, in heavens name, think different? Nijdam (talk) 10:02, 15 June 2010 (UTC)

My question was how do you know P(C=1|X=1, H=2 OR H=3})=P(C=1|X=1)=P(C=1)? Martin Hogbin (talk) 10:54, 15 June 2010 (UTC)

Simple calculation. The simpliest: {H=2 or H=3} is part of {X=1}, hence forms no further comdition. Nijdam (talk) 13:04, 15 June 2010 (UTC)

Suppose the player picks door 1, the host (who chooses a goat door at random) opens either door 2 or door 3 but we do not know which and the player then calculates the odds of winning by switching. We all agree that this is simply calculated as 2/3. The player is just about to ask for the swap when they are told that the host has actually opened door 3. On what basis do they revise their calculation of the probability of winning by switching? Martin Hogbin (talk) 22:37, 15 June 2010 (UTC)

Why ask me? You perfectly know yourself! Before this information comes to the player, she decides on the basis of P(C=1|X=1, H=2 OR H=3). With the new info she calculates: P(C=1|(X=1,H=3).Nijdam (talk) 07:55, 16 June 2010 (UTC)

But she has no need to do the calculation from scratch again. It is perfectly obvious that, if the host has opened a goat door uniformly at random, the probability P(C=1) does not change when the door number is revealed. The P(C=1) is obviously independent of P(H=3) in this case.

You were careful to explain that probability is a state of knowledge, which I have always agreed. What new information about P(C=1) is revealed by a randomly chosen door number. Martin Hogbin (talk) 08:45, 16 June 2010 (UTC)

(editing2)

I won't deny P(C=1) does not change. P(C=1)=1/3, and will keep its value till infinity. I'll have to refrase your second remark. I think you mean to say: the events {C=1} and {H=3} are independent. They are, but not because this is a given fact in the problem formulation. It has to be proven. About your last remark: as I mentioned: P(C=1)=1/3, there is nothing more to be known about it, and nothing can change it. So: what do you mean? Nijdam (talk) 15:31, 16 June 2010 (UTC)

We're going round in circles. After you may have answered the above question, respond to the following:

What I meant to say was that (with X=1 assumed throughout) when it is revealed that H=3 it is obvious that P(C=1|H=3) = P(C=1|H=3 OR H=2)

I maintain that it is obvious that P(C=1|X=1,H=3) = P(C=1|X=1,H=3 OR H=2) = P(C=1) from symmetry.

It depends on what you mean by obvious. Indeed do they have the same value, but it concerns different types of probability, and that they have the same value is not a given fact in the problem formulation (see below).

Do you agree that from symmetry P(C=1|X=1,H=3) = P(C=1|X=1,H=2)?

Of course, no need to show me the rest of the derivation. With this you do show already the necessity of some arguing, reasoning or proof. Nijdam (talk) 05:54, 17 June 2010 (UTC)

IF the player has chosen X=1 and the host has opened H=3, then the player uses this information by basing her decision on: P(C=2|X=1,H=3)

Nijdam (talk) 18:20, 16 June 2010 (UTC)

Yes I agree all that.

Which door has been chosen and which one opened?Nijdam (talk) 05:54, 17 June 2010 (UTC)

Door 1 has been chosen and door 3 opened.

Well, you beat me. How on earth can you still defend the simple solution as being correct??????? Nijdam (talk) 05:12, 18 June 2010 (UTC)

Maybe you misunderstand what I am claiming. I do not say that conditional solutions such as you propose are incorrect, you are, of course, quite correct in all your arguments. What I say is that in the symmetrical case some of the facts are obvious and do not need the formalism of conditional probability to arrive at.

Sorry Martin, but this sounds quite illogic. I asked you why you defend the simple solution to be correct, and your answer is I might think you consider conditional solutions incorrect????

My point still is: all simple solutions are incorrect!

Incorrect in what way? They give the correct answer within their range of applicability (the symmetrical case only). They only fail if used in a case where the host chooses a goat door non uniformly.

Incorrect as they do not calculate the right appropriate probability. This has nothing to do with the symmetry in the problem.

I agree that the simple solutions fail to distinguish between the possible doors opened by the host, however, most people do not even think of the possibility that the door opened by the host might make a difference. In some cases it does but in the symmetrical case our natural intuition that it does not matter which door the host opens turns out to be valid.

It is not a matter of what most people think. Why should it. Most people think the chances are even. And again: in the symmetrical case your intuition apparently fails; it just matters that a door has been opened.

When I say that P(C=1|X=1,H=3) = P(C=1|X=1,H=3 OR H=2) = P(C=1) from symmetry, I do not meant that I take the unconditional sample set and, using symmetry, condition it to obtain P(C=1|X=1,H=3).

Do you at least calculate P(C=1|X=1,H=3)?.

I calculate the probability that the car is behind door 1 given that the player has chosen door 1 and the host has opened an unknown one of door 2 and 3. I then assert that knowing that host has opened door 3 does not change this probability, thus I calculate P(C=1|X=1,H=3).

There you are: it shows that P(C=1|X=1,H=3) has the same value if you like as the probability of throwing 1 or 2 with a fair die.

I am using the notation P(C=1|X=1,H=3) purely because you asked me to. I would prefer to put it in words, less formally, and say the probability that the car is behind door 1 after the host has opened door 3 is obviously the same as the probability that the car is behind door 1 after the host has opened an unknown one of either door 2 or door 3 and this is obviously the same as the probability with which the car was originally placed behind door 1.

Well if you want to use the exact wording, like here, it makes no difference. And also then you have to realize they are different probabilities. Or do you intend to use vague wording in order to get around this?.

I think it is best to use slightly vague wording at the start of the article because most people do not consider the possibility that it might matter which goat door is opened by the host and, in fact, it turns out that it does not matter. I accept that most people (at least consciously) do not go through my argument in their head. They just ignore the fact that a specific door is opened. In the non-symmetrical formulation, this thinking would lead to the wrong answer.

You're running around in circles, without approaching the desired goal.

I do not assert that the simple solution is the best solution, especially for teaching probability. The MHP can be used as an excellent example of the way that a seemingly unimportant fact can significantly alter a conditional probability. I have no desire to hide this fact in the article, in fact I would welcome more discussion of the subject, but in the appropriate part of the article.

Better would be to assert it is no solution at all.

On the other hand I do not accept that it is good practice to use an inconsistent approach to the subject (as Morgan do) to demonstrate the pitfalls of ignoring information presented. Had Morgan added, 'assuming all we know is that the cars were placed uniformly at random at the start', and 'using only information given in the problem statement' I would have much less problem with their paper.

You're constantly missing the point.

We too must be careful not to use unfair tricks to make our conditional probability exposition more exciting. In the case that the host chooses a random goat door, the simple solution does give the correct answer, because of an obvious and intuitive symmetry. Whether that makes it strictly 'correct' or not we could argue forever.Martin Hogbin (talk) 09:20, 19 June 2010 (UTC)

As you've seen above: I can think of millions ways to get the "correct" answer (as being the value 1/3).

Maybe you consider this not to be obvious but I think that many people do, and it happens to be true. Can we not agree therefore that informally for the symmetrical conditional case, the unconditional solution (in which an unknown one of door 2 and door 3 is opened) suffices? Martin Hogbin (talk) 09:20, 18 June 2010 (UTC)

Indeed, as you say, many people do not consider this to be obvious. The more or less "obvious" point is only the fact the different probabilities share the same value! It is however important to know they yet are different!Nijdam (talk) 07:37, 19 June 2010 (UTC)

Yes, of course we should explain your point in the appropriate section of the article but most people will not be interested.

We should make this clear direct from the start.

It look as though we will have to agree to disagree. You are insistent that your preferred solution is the only correct one. It is most unusual in mathematics for there to be only one correct way of solving a given problem. The simple solution gives the correct answer to the symmetrical problem because of an obvious and intuitive symmetry which makes the conditional and unconditional solutions identical. I have tried hard to find some middle ground on which we could agree, such a calling the simple solution 'informal', but you are having none of it. You insist that your view and method are the only correct ones.Martin Hogbin (talk) 10:12, 20 June 2010 (UTC)

There are many ways to solve the problem. I thought we agreed on this. But solving means calculating the conditional, or if you wish, posterior probability. In what way this is done is unimportant, as long of course it is done correctly. The use of the symmetry is one method. But that's something completely different than the simple solution. Do not mix things up. Perhaps the only point of disagreement left is: you take a different formulation to be the MHP than me. I do not understand why. The only reason I can think of is that for this (simplified) version, the simple solution applies. To be specific: in my MHP the host has opened the door before offering to switch, in your version the host offers to switch before it is known which door he will open. Am I correct?Nijdam (talk) 10:38, 20 June 2010 (UTC)

No. Whether problem is conditional or not is not dependent on the time that the question was asked, it depends only on whether it is the intent of the questionner that a condition should be applied.

For example, if 'the host offers to switch before it is known which door he will open' this does not necessarily mean that the problem is unconditional. It might be known that the host will open a specific door.

Alternatively, it may be that, 'the host has opened the door before offering to switch' but the person asking the question has not seen which door or has been opened, or has seen the door the door but does not wish that information to be used in calculating the required probability.

The only thing that determines whether a problem is conditional or not is the intent ofthe questionner. How this is expressed in the problem statement depends on the context.

In a statistics exam, for example, the person setting the question must make clear exactly what is given in the problem statement and what is not.

In a question to a popular magazine, however, it is the duty of the person answering the question to decide exactly what it is that the person asking the question really wants to know. In the case of Whitaker's question, vos Savand made a good job of this.

It it almost certain that Whitaker did not want his question to apply only to the specific doors mentioned. It is, in my opinion, very likely that he just wanted to know whether it was, in general, better to swap or not, and by how much. In other words the answer to the unconditional formulation. Martin Hogbin (talk) 08:56, 23 August 2010 (UTC)

Nijdam, you have not responded to the above points. Martin Hogbin (talk) 14:20, 3 January 2011 (UTC)

Well, it means repeating what has been said several times. Also Whitaker describes a situation where a door has been chosen and one being opened, It is the what makes the MHP an intriguing problem, because the player (reader) is confronted with two closed doors instead of three, and hence most people are inclined to think the odds are equal. Remember the furious dispute with the famous Erdoes, who only very reluctant came to understand different. The odds may be equal under special assumptions, but not on the base of the argument that only two doors are left and implicitly assume they have equal probabilities. Be assured that the player knows which door she has chosen (!) and which one is opened by the host. It doesn't matter whether these are doors 1 and 3, any other combination will do. Of course one may consider various other forms of a kind of MHP, interesting stuff to be placed in the rest of the article for the experienced reader. Nijdam (talk) 23:20, 3 January 2011 (UTC)

Insight

You have to come to the insight, that:

1. in the MHP the host first opens a door and then offers to switch.
2. the decision has to be based on the conditional probability given the number of the opened door.
3. symmetry may be helpful in calculating this probability, but does not make it possible to avoid this
4. although the unconditional probability and the conditional one for the chosen door to hide the car are both 1/3, they are different items
5. a solution is incorrect if this distinction is not made

Nijdam (talk) 10:44, 19 June 2010 (UTC)

It is of course possible to formulate a slightly different problem, in which the host announces his intention to open one of the other doors, and before actually opening that door, offers the player to switch to the door he will keep closed. For this formulation the simple solutions are correct. However, this is not what is considered the MHP. Read the formulation, look at the pictures people draw. Nijdam (talk) 10:00, 20 June 2010 (UTC)

Memo

Difference between: "behind one of these doors is a goat" and "behind this door is a goat".Nijdam (talk) 09:52, 15 June 2010 (UTC)

Analyse

Formulation: A: specific doors B: in general

A: example X=1 and H=3
Solution:
calculate P(C=2|X=1 and H=3)
simple "solution" fails, likewise equivalent solutions
many doors solution fails
most simulations are wrong

B: factually is the player asked whether to switch or not even before picking a door
seems contradictory to the formulation: ...host opens door and then offers...
simple solutions

Issues

Some assumptions do not influence the discussion, nor the problem:

• Car is placed randomly
• Choice of door is independent of the place of the car
• Host chooses randomly, if necessarily.

We have 2 main interpretations of the MHP (F1 and F2), and 2 main solutions (S1 and S2).

F1: the player is offered to switch before she picks a door. As the player will know which door she will choose, the audience has to decide.

F2: the player is asked to switch after the host opened a door. The player decides or the audience. As all are aware of the rules, there is no difference.

S1: simple solution: "the car is with probability 1/3 behind the chosen door, hence with probability 2/3 behind the remaining one"

S2: based on conditional probability given the picked door and the opened door

F1 is solved by S1 (??? questionable: it may be better to base the decision on the different conditional probabilities)

F2 is solved by S2

F1 is not solved by S2

F2 is not solved by S1

My opinion: F2 is the "right" interpretation

Some people wrongly defend: F2 is solved by S1.

It is a misunderstanding that S1 is correct to solve F2 because of the symmetry. Although symmetry may be used to show that the conditional probability is equal to the unconditional, S1 does not mention this, and hence fails as a solution to F2.

Nijdam (talk) 20:51, 10 October 2010 (UTC)

Switch

"Is it in your advantage to switch?"

The paradox lies in many people saying: "It doesn't matter, as the odds are equal for the two doors." And they may be right(?), as with a certain strategy of the host, the odds indeed are 50-50. The numerical answer is indeed correct, but the reasoning behind it is not. Are they correct?

With certain weak assumptions it is indeed advantageous to switch. Is it enough to answer the question by saying: "Yes"? Of course we have to know what the answer is based on.

The overall chance of getting the car by switching is 2/3. Is this sufficient to motivate the switch? No, in the particular situation of the player, the chance may (theoretically) be less than 1/2. So the arguing must extend to conditional probabilities.

No, there is no reasonable formulation of the problem (which is soluble) in which the answer is not 2/3. Martin Hogbin (talk) 09:15, 23 August 2010 (UTC)

Martin, Nijdam: see [conditions/formulations] Regards, Gerhardvalentin (talk) 14:55, 23 August 2010 (UTC)

Yes Martin, there are: i.e. the ignorant host, the lazy host. The ignorant host opens any door at random except the chosen one: P(C=2|X=1,H=3)=P(H=3|X=1,C=2)P(C=2|X=1)/P(H=3|X=1)=[1/2.1/3]/[1/2.1/3+1/2.1/3]=1/2. Lazy host opens if possible door 3: P(C=2|X=1,H=3)=P(H=3|X=1,C=2)P(C=2|X=1)/P(H=3|X=1)=[1.1/3]/[1.1/3+1.1/3]=1/2. Nijdam (talk)

These are not reasonable formulations of the problem. Please give me any reason, mathematical, logical, philosophical, or based on real-world knowledge, to take the initial car placement as uniform but the host door choice as non-uniform. Martin Hogbin (talk) 21:00, 24 August 2010 (UTC)

Wrong simple solution

(Copied from User_talk:Martin Hogbin)Nijdam (talk) 09:50, 8 January 2011 (UTC)

And Richard, I really am fed up by having to repeat this over and over, without getting an answer. So once and for all, answer to the following, and restrain yourself to what I say. One form of the simple solution reads: Due to the random placement the car is with probability 1/3 behind the chosen door 1. As the opened door 3 shows a goat, this door has clearly probability 0 to hide the car. Hence the remaining door 2 must have probability 2/3 on the car. it is this formulation I'm fighting. Especially because it looks quite plausible and is copied by lots of school pupils and students who gets assignments about the popular MHP. Now just comment on what I wrote here. Nijdam (talk) 22:06, 7 January 2011 (UTC)

@Nijdam, you are absolutely right. "Hence" is wrong. What is said is a nonsequitur. The fact of specifically Door 3 being opened could change the likelihood that the car was originally hidden behind Door 1. So far we only used "all doors initially equally likely" and that isn't enough to get a conditional result, while this argument is about a conditional result, since it talks about how probabilities change (or don't change) on getting the "Door 3" information. There is a missing step where "no host bias" has to be explicitly used. Eg, appeal to independence: by symmetry, the number of the door opened doesn't change the probability the car is behind door 1. Or if you prefer, use Bayes' rule and show by explicit computation that the odds on Door 1 hiding the car isn't changed by the information *which* specific door is opened. Next task, find a reliable source which explicitly makes this point, because I'm afraid that some editors could consider this observation of yours "OR". (Or write it yourself: why not submit a small note to Statistica Neerlandica?) Richard Gill (talk) 07:24, 8 January 2011 (UTC)

Simple explanation (add)

The conclusion of the simple solution is that "always switching" beats "always staying": overall success-chance 2/3 versus 1/3, meaning that on the average 2/3 of all players will win the car. It is not clear however whether this result also holds for the player in the situation given in the problem. Some sources are therefore interested in something different: given that the contestant has chosen Door 1 and given that the host has opened Door 3, revealing a goat, what is now the probability that the car is behind Door 2? The reason for this more subtle and detailed inspection is that in principle it is possible that in some of the six possible situations (specific door chosen by guest, specific door opened by host) it would be better to switch and in other situations better to stay. If so, the overall succes-rate could be made better still than 2/3, the success-rate which is achieved by *always* switching.

The error lies in the term switching. Switching in general is not the same as switching in the situation of the player. The player can only switch to door 2 (the remaining door). She doesn't have the option to switch to door 3 with the goat, whilst in general she may. Unconditional the probability for every door to hide the car is 1/3.

Issue

Versions

Although I'm not sure whether this is the appropriate place, I take the opportunity to explain about the versions of the MHP and the solutions.

Conditional formulation (F0)

A player is standing on stage in front of three doors, numbered 1, 2 and 3. A car has been placed behind one of these doors, and the player may have what is behind the door of her choice. She points to door 1 as her first choice and before she opens it, door 3 is opened by the host, showing a goat. Then the host offers the player to change her choice. It is common to assume that the car is placed randomly, the choice of the player is independent of the position of the car, the host always shows a goat and when the host has a choice he chooses randomly.

Unconditional formulation (F1)

A player is standing on stage in front of three doors, numbered 1, 2 and 3. A car has been placed behind one of these doors, and the player may have what is behind the door of her choice. The rules of the game are explained to her, i.e. She may choose a door and have what is behind it, and before she will open it, the host will open one of the two remaining doors, showing a goat and offering her to change her decision It is common to assume that the car is placed randomly. Before the player makes her first choice, the audience, unaware of the position of the car, is asked whether the player will have to stick to her first choice or must switch.

Conditional solution (S0)

The player calculates the conditional probability that the car is behind door 1, the door of her first choice. It turns out to be 1/3. Alternatively she may calculate the conditional probability that the car is behind door 2. It turns out to be 2/3. (As the conditional probability for door 3 is 0, the conditional probabilities for door 1 and door 2 are of course complementary.)

Unconditional solution (S1)

As the car is placed randomly the car is with probability 1/3 behind the door the player will choose. Hence on the average by switching a player will get the car 2/3 of the times. This solution is also referred to as the simple solution.

Simple conditional solution (S2)

Due to the obvious symmetry in the problem (NB we have to explain the standard assumptions before), the probability for the chosen door 1 to hide the car has the same value 1/3 before the player makes her choice and after the host has opened door 3. Hence at the moment of decision the remaining door 2 has probability 1-1/3=2/3 to hide the car. Nijdam (talk) 12:21, 4 January 2011 (UTC)

Fact is that F0 is solved by S0, and F1 is solved by S1. And what is important: F0 is not solved by S1, and it is a serious logical error to consider S1 to be a solution to F0. Yet several sources, although mostly the non-academic, more popular ones, and also some participants in the discussion, want the article to at least start stating S1 as a simple but sound solution to F0, and not allow other editors to immediately mention the criticism on it.

NB.: I'm not sure anymore whether F1 is indeed properly solved by S1. The audience could better reason: In any of the situations where door No. x has been chosen, and door No. h has been opened, the relevant conditional probability is 2/3. Hence switching will be optimal in any situation.

I suggested a kind of compromise (User_talk:Martin Hogbin#New Example, subsection Acceptable formulatio, what seemingly has been moved by Hogbin to User_talk:Martin Hogbin#Monty Hall discussion,), in introducing solution S2, which may be called "simple conditional solution". It is a correct solution (explanation) of F0. It shows in a more intuitive manner, why the required conditional probability is 2/3. But id didn't lead to the desired consensus.

Words

"switching gives the car when initial pick is a goat"

Is this a solution?

the player has picked a door! and switching will give the car if his picked door hides a goat. That's something different

X=1, switching gives the car if C<>1.

X=1, H=3, switching gives the car if C=2.

To switch or not to switch

The question is: Should the contestant switch? Under the usual assumptions the answer is Yes. Is this satisfactory as a solution to the MHP? Maybe, but mostly, as not always, the reply will be: Why? If the contestant has adopted the strategy of always switching, the correct answer is: Because she always will switch. Followed by: Why does she follow this strategy? Answer: A contestant with this strategy gets the car 2/3 of the times. Which times? In repeating the game. Is this strategy optimal for the contestant? She is in a specific situation, having chosen door No. x and seeing door No. h opened. We cannot tell, unless we further investigate the problem. Then we will discover that all the relevant conditional probabilities will be equal and hence equal to the unconditional one. Only then may we conclude that always switching is optimal. But notice, this conclusion is based on the conditional probabilities.

Distinguishable doors

Sometimes people argue that the doors are indistinguishable. This is of course nonsense, as the doors are clearly visible to anyone. Perhaps one means to say that the door numbers are not relevant. Then we have to ask: not relevant to what? We can say that door 1 has been chosen and door 3 opened, but these numbers are indeed not relevant. instead of doors 1 and 3, we might have chosen doors 3 and 2 for instance. Or from the irrelevance of the door numbers we may conclude about the symmetry and use this to calculate the conditional probabilities.

Independence of choice and car position

Most authors forget to mention as an assumption that the initial choice of door will be independent of the position of the car.

Chosen door

It will be very difficult for people to understand the difference between:

• The chosen door is No. 1, the car is behind door 1 with probability 1/3.
• The car is with probability 1/3 behind the chosen door
• Given door 1 has been chosen, the car is with (conditional) probability 1/3 behind door 1 (or the chosen door)

Game theory

If the MHP is treated as a game theoretic problem, the host should be able to determine his strategy, or else there is just one single player.

And yes Nijdam, the host indeed is able to determine his own strategy just as he should like. That's completely up to him. There is no "or else ...".

Moreover, it is completely irrelevant whether the candidate knows about such strategy, or not.

Please note that any host's strategy is completely irrelevant for the decision asked for.

All sources agree that staying never can be of advantage for the candidate and never can be the "better" decision.

So any thought that considers "staying" is dominated and can be discarded just from the outset, as game theory can show and already did show.  Regards, Gerhardvalentin (talk) 08:05, 14 September 2011 (UTC)

What's your point? Nijdam (talk) 08:18, 14 September 2011 (UTC)

In the solution using Game Theory, the host indeed is able to determine his own strategy just as he should like. That's completely up to him. There is no "or else ...".  Gerhardvalentin (talk) 09:26, 14 September 2011 (UTC)

Devlin

Might it be that what is obvious and what needs argument depends on who you are? It's a fact that some people are unhappy with Devlin's argument, including Devlin himself, and Devlin himself did not know how to explain what he was doing. Let me also point out a mathematical fact. Consider statement (A): "the chance is 2/3 that switching will give you the car, whatever door you first chose and whatever door is opened by the host". Note: statement (A) specifies 6 different conditional probabilities: Prob(car at a|player chose b, host opened c)=2/3 for all the 6 permutations (a,b,c) of (1,2,3). Consider also statement (B) "the initial location of the car is completely random" and statement (C): "the choice of the host is completely random (if he has a choice)". It's an inescapable mathematical fact that (A) is true if and only if (B) and (C) are both true. Somebody who claims (A) has explicitly or implicitly assumed and used both (B) and (C). Someone who claims (A) and uses (B) to support his claim but not (C) has not given a correct argument. This is in essence what Devlin did. Devlin is a mathematician and a figurehead, a leading public figure, in the mathematical community. By his own criteria and the criteria of the community he was writing for, his argument was wrong. I don't ask you to find all this important. I just hope that intelligent lay-persons could at least appreciate the way mathematicians think. Society uses mathematics all over, and mathematics contributes vastly to society. Our value to science and society is precisely that we care about these issues. MHP on wikipedia is a topic which is interesting both to lay-persons and to people learning mathematics. Since it is a fixed item in every introductory probability and statistics course, this means that a huge proportion of the people consulting the wikipedia page on MHP will actually be students of probability or statistics. I wouldn't even be surprised if it's a majority. Like it or not, laypersons and mathematicians have got to work together on this article. There always have been and always will be roughly equal numbers of sources, editors, and readers of the two categories. Richard Gill (talk) 08:25, 7 November 2012 (UTC)

Gill

I don't want to make a meal of the door identity question. I just want any arguments which are presented as logical arguments to be correct. Devlin pretends to present a logical argument but gives no justification for his last step. Some people thought it was obvious, some people thought it wasn't obvious. When Devlin was informed of this he suddenly began to doubt, himself. In order to justify Devlin's last step you need to explain why the question of which particular door he opens (door 2 or 3) is irrelevant to the question of whether or not the car is behind door 1. You can say it is obvious, or you can write out the argument if you think it is not obvious. If you say something is obvious, but you are challenged (as Devlin was) you are obliged to fill in the details. This is not pedantry. After all, by opening door 2 or door 3 he does dramatically change the chance the car is behind door 2 or behind door 3! Some readers are uneasy here. Devlin was uneasy. Nijdam screams murder. The more you put in the more you get out. I don't think it's a big deal. But let's at least be clear as far as the logic is concerned. If you want to argue that switching gives the car with probability 2/3, then you have a very easy job. The simple solution. Switching gives the car if and only if your initial choice is a goat, probability 2/3. Combined doors: it's as if you are offered the option of exchanging door 1 for doors 2+3. 1/3 versus 2/3. Very intuitive, completely true. If you want to argue that switching gives the car with probability 2/3, given you chose door 1 and the host opened door 3, you have a little more work to do. You'll need somewhere to use your (natural) assumption that the host is equally likely to open either door if he has a choice. For instance, you could do the simple solution first and then say that by symmetry the conditional probabilities are the same. I think the whole idea that some solutions are correct and others incorrect is a waste of time. Every solution is correct in its own terms. Let's just be clear about this. Secondly, like it or not, almost everyone who is trained in elementary probability, and certainly any beginner in this field, is going to sit down and calculate the conditional probability of finding a car behind door 2 given the player chose door 1 and the host opened door 3 from first principles, like Sevlin did in his second paper back in 1975. Like it or not, the article is going to contain a heap of material on conditional solutions. On wikipedia we are not supposed to make original syntheses, but my job as a mathematician outside of wikipedia is to do this. I like to show people the synergy between simple and conditional, the easy bridges, the smart ways to avoid doing blind computations. I would also like to counter the Morgan et al. propaganda! Richard Gill (talk) 15:06, 6 November 2012 (UTC)

You misunderstand my position. The fact is that different solutions get different conclusions under different assumptions. The numbers 2/3 may be the same but what it means is different. 2/3 of what? My opinion is that it is up to the consumer to choose. The simple solutions and the conditional solutions talk about different things. The decision theoretic solutions yet again. If you want to deduce the probability of winning by switching given you chose door 1 and the host opened door 3 you have more work to do and need to make more assumptions than if you want to deduce the overall probability of winning by switching. I do not say you have to go for one or for the other. I say you legitimately can go for one or the other. The consumer does need to be correctly informed. And false logic has to be exposed. If your solution is implicitly relying on a fair host assumption but you do not use it explicitly, then your logic is sloppy, your argument, at best, in incomplete. Thesimple solutions don't use this assumption and don't need it either. They draw a weaker conclusion than the conditional solutions. These are mathematical facts of life, like it or not. Richard Gill (talk) 07:16, 3 December 2012 (UTC)

Contact

Beste heer Nijdam,

Graag zou ik contact met u onderhouden over een aantal pagina's gevonden op Wikipedia (eng). Mijn email adres is: cwvugs@ziggo.nl

Met vriendelijke groet,

Kees Vugs — Preceding unsigned comment added by C.W. Vugs (talk • contribs) 15:49, 23 December 2012 (UTC)