Wikipedia:Reference desk/Archives/Mathematics/2009 May 31

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May 31

Lottery question

Does someone who has played the lottery for most their adult lives have a better chance at winning the lottery than someone who has never played? If the lottery veteran and rookie each put $5 on tonights drawing, does the veteran have any kind of edge over the rookie? --67.85.117.190 (talk) 00:37, 31 May 2009 (UTC)

No. Dauto (talk) 00:53, 31 May 2009 (UTC)

Someone who has played for many years has the same chance at winning the *next* lottery as someone who has never played. Someone who has played for many years has a greater chance at winning *at some point over his life* that someone who plays only once. However, "win" is misleading. Each lottery ticket has a negative expected value, so the person who never plays actually faces a greater expected return than someone who plays once. Someone who plays once, in turn, has a greater expected return than someone who plays a lot. Wikiant (talk) 00:56, 31 May 2009 (UTC)

OP asked two questions. The short answer, "no" is precisely correct for the second question, where the same wager on the same night is made by a veteran and a rookie. However, the more open-ended first question requires a slightly more detailed answer, and for some lotteries, the answer could very well be that the veteran could have an edge. In some lotteries in US states, if a winning number has not been sold, the amount is rolled over into the next weeks lottery. This happens often, and the expected winnings change in the next drawing. A veteran might vary the amount bet to reflect the different odds. They will still expect to lose in most situations, but will have slightly better odds than a rookie, who is not likely to know this fact. In rare circumstances, the winning amount can accumulate to the point that the expected winnings exceeds the ticket price.Sphilbrick (talk) 00:07, 1 June 2009 (UTC)

Because of rollover jackpots as Sphilbrick describes, the expected winning for a ticket in a given drawing can be negative (the usual case) or positive (sometimes) depending on the amount already in the pot. However, even when the expected winning is positive, for most people (those without millions of dollars in the bank already), buying a ticket is a bad play according to the Kelly criterion. See [1] for explanation. 67.122.209.126 (talk) 00:19, 1 June 2009 (UTC)

Dauto's simple answer, "no", was correct. It could have been more correct, had it been "no and no." Each dollar played by the rookie player has the same chance of winning (and probability of loosing) as the dollar played by the veteran player. It doesn't matter if the veteran player played more, less, or not at all in prior drawings. Those other plays are not a factor. Assertions by a veteran player to the contrary are likely the result of a belief in the law of averages falicy or perhaps the gambler's fallacy. -- Tcncv (talk) 01:15, 1 June 2009 (UTC)

Dauto's simple answer was correct in the case of the second question. If the second question was intended to be a restatement of the first, it applies to what the OP was thinking, however, the first question as asked does not preclude the possibility of rollover lotteries (which exist in the real world) and the possibility that a veteran will abstain in lower payout situations. The second question precluded that possibility - but the general answer is that a veteran can have a different expect payout than a rookie because a veteran may abstain in situations where a rookie may choose to play.Sphilbrick (talk) 14:41, 1 June 2009 (UTC)

While strictly true, the only correct sized bet from a statistical standpoint in virtually any lottery is to bet nothing. I think we can assumed from the OPs question that the veteran regularly plays the lottery, rather than just regularly checking the expected prize fund and number of ticket sales and buying a ticket on the extremely rare occasions (if ever) that there is a positive expectation. --Tango (talk) 22:46, 1 June 2009 (UTC)

Strictly speaking, "no and no" is incorrect. It's only correct if you view the second question as a rephrasing/clarification of the first question, in which case there is no need to repeat yourself. (I agree with the answer of "no" for the second question, for the reasons already given.) If you feel the two questions need two distinct answers, the answer to the first question as-written is a qualified "yes" - a person who has never played the lottery has a zero chance of winning any of the games he has never played (barring freak events like picking up a winning ticket off the street, which factor into each side equally), so the chance of a habitual player winning at some time in their life is greater. Infinitesimal, but still greater than zero. It still can be a "yes" if we permit the rookie to play one game, and define winning as "making a net profit from the lottery (prize money minus ticket costs)" - stick with me here. Say we have a hypothetical lottery with a $1 buy-in, a million dollar payout, and a chance of winning of 1 in 2 million (50% payout). A single $1 player has a 1,999,999/2,000,000 chance of a net loss of $1 and a 1/2,000,000 probability of net positive gains ($999,999). A habitual player who plays $1/day for ~28 years (10,000 tickets) has a ~1,990,025/2,000,000 chance of losing $10,000, and a ~9,975/2,000,000 chance of getting at least one winning ticket (which would put him at a net positive). While the fractional expectation value is the same (50% loss on money wagered), such that the habitual gambler is expected to lose more overall ($5,000 vs. $0.50), the chance that the habitual gambler will net positive over his lifetime is greater, which is offset by the increased amount which he loses, in the very likely case he loses every drawing. -- 128.104.112.106 (talk) 23:30, 1 June 2009 (UTC)

Parentheses and functions

I'm looking for some clarity as to what is best form with regard to function notation, in the real world and in writing here on Wikipedia. Sometimes, one sees $\cos x$ or $\ln x$, and at other times, $\cos(x)$ or $\ln(x)$. Which is preferred? Is one more formal than the other? Is one more correct than the other? I've always been inclined to use the (), only because $\cos$ is, indeed, a function. Is it simply a matter of laziness or laxness in writing? Thanks for any help. —Anonymous Dissident^Talk 12:30, 31 May 2009 (UTC)

The problem with omitting the parentheses is that it becomes ambiguous exactly what expression one is supposed to be applying the function to. For example, what does the following expression mean?

y = cos x + 2

Is it the cosine function shifted up by 2, or is it the cosine function shifted to the right by 2? With the parentheses, one can be sure exactly which one we're talking about.

So I think the answer is, yes, it is simply a matter of laziness/laxness in writing. --COVIZAPIBETEFOKY (talk) 13:18, 31 May 2009 (UTC)

Should be 'shifted to the left'. -- Meni Rosenfeld (talk) 19:59, 31 May 2009 (UTC)

You're right. I thought I checked that... --COVIZAPIBETEFOKY (talk) 21:26, 31 May 2009 (UTC)

I'd say it's for brevity or compactness in notation, rather than laziness. There are cultural conventions at work about the order of operations which allow suppression of parentheses in unambiguous ways. In almost all contexts, parentheses in cos(x) are just as unnecessary as in (x)². Staecker (talk) 13:23, 31 May 2009 (UTC)

This is actually covered in the Manual of Style, see MOS:MATH#Choice of type style. SpinningSpark 13:58, 31 May 2009 (UTC)

It is quite common convention that multiplication goes before functions, and functions before addition, so $\cos 2x+y\equiv (\cos(2x))+y.$ This is quite useful because many important identities, esp. in trigonometry, include functions of doubled or halved argument, so the convention simplifies notation, It works same way, as omitting parenteses in multiplication vs addition simplifies eg. polynomials (try to write a third degree polynomial $Ax^{3}+Bx^{2}+Cx^{2}+Dx+E\,\!$ with full parentesising...) --CiaPan (talk) 17:17, 31 May 2009 (UTC)

In informal contexts, the decision to parenthesize or not is pretty random. For example, looking at a whiteboard I had recently used, I found I had written:

$(1+\epsilon )^{\ln n/\ln 2}=e^{\ln(1+\epsilon )\ln(n)/\ln 2}=n^{\ln(1+\epsilon )/\ln 2}$.

Whereas, in more formal contexts, there is some subtlety to the choice of using parentheses; the goal is to make the expression as easy to read as possible. Parentheses are used if clarity is needed, for consistency, or for emphasis; parentheses are sometimes omitted if obvious, and the function is being used in a heavily nested expression where you already have lots of parentheses (so that adding more just makes it harder to read). One convention is that (sometimes) a function f might use parentheses if applied to a single element, but omit parentheses if applied to a set of elements. This convention is used in particular for the norm function and for field isomorphisms; for example, I once wrote up a problem set containing the following:

$\tau (a)-\tau (b)=\tau (a-b)\in \tau Q^{k}$

Parentheses are used for $\tau (a-b)\,$ because I have to; while it would have been stylistically ok to write $\tau a-\tau b\,$, for consistency I included parentheses there, too. Whereas parentheses are omitted in $\tau (Q^{k})$ because $Q^{k}$ is a set instead of an element. Eric. 131.215.159.116 (talk) 22:12, 31 May 2009 (UTC)

What I never understood was the inconsistency in that $\cos ^{2}(x)$ is the cosine of x, squared, but $\cos ^{-1}(x)$ is the inverse cosine of x, as opposed to one over the cosine of x. I guess this is why I write $\arcsin(x)$ instead. Readro (talk) 15:41, 1 June 2009 (UTC)

$\cos ^{2}(x)$ is an abuse of notation. (cos(x))² is far more common than cos(cos(x)), so it gets the shorter notation, even though it is inconsistent with the usual meaning of such notation. It is universally used, though, so there is no ambiguity. It's notation like $\cos ^{-2}(x)$ you need to be worried about! (Never use notation like that, you're just asking for trouble.) --Tango (talk) 15:56, 1 June 2009 (UTC)

Mathematical Drawing

For a paper I am writing, I need to include some mathematical drawings such as triangles and arrows (with lines of different styles, dashed for example). Unfortunately, I am using Windows so I am looking for something analogous to xFig in Linux for Windows. I found WinFig but it is shareware so it limits me unless I pay them. So, does anybody know a program like xFig for windows which is free. I am not doing anything complicated, just a lot of lines, triangles, and arrows of different styles. Thanks! -Looking for Wisdom and Insight! (talk) 22:25, 31 May 2009 (UTC)

You might try Inkscape. --Thinboy00 @031, i.e. 23:44, 31 May 2009 (UTC)

If you are using TeX, there are various macro packages intended for typesetting commutative diagrams. Maybe those could help in a simpler way than an external drawing system. 67.122.209.126 (talk) 01:37, 1 June 2009 (UTC)

Actually, I am using PDFTex but there are a lot of objects I am drawing so it gets annoying programming them in Latex. A program like xFig is much much faster and then I simply export the image as EPS and put it in my document with no problems. That is why I prefer an external program (which can also export to EPS preferably).-Looking for Wisdom and Insight! (talk) 02:34, 1 June 2009 (UTC)

In that case, I would recommend Ipe, which allows for embedded LaTeX and can export to PDF or EPS as well as its own format. Michael Slone (talk) 03:23, 1 June 2009 (UTC)

Check out OpenOffice.org Draw, which too allows you to export drawings as eps. Alternatively you can install cygwin on windows and then install xfig. Takes an hour of installation + configuration, but is worth the time if you are already comfortable with xfig. See specific installations instructions at this site. 98.220.252.228 (talk) 03:51, 1 June 2009 (UTC)

With pdfTeX, I've used MetaPost and PGF/TikZ. TikZ is more expressive, I feel, though I tend to use MetaPost more often. These are text-based systems in which you more or less "program" your drawings, which can be very nice sometimes but occasionally is frustrating. For simple drawings, though, they aren't too bad. —Bkell (talk) 04:07, 1 June 2009 (UTC)

I sometimes find myself writing eps files manually in a text editor. The PostScript language is not terribly difficult to learn, I don't think it's any harder than MetaPost. — Emil J. 13:16, 1 June 2009 (UTC)