13 What is the answer to the Monty Hall, Envelope, or Birthday problem?

There is a classic probability puzzle, which is called the Monty Hall problem. Here's a nice description from the rec.puzzles FAQ. "The Monty Hall problem can be stated as follows: A gameshow host displays three closed doors. Behind one of the doors is a car. The other two doors have goats behind them. You are then asked to choose a door. After you have made your choice, one of the remaining two doors is then opened by the host (who knows what's behind the doors), revealing a goat. Will switching your initial guess to the remaining door increase your chances of guessing the door with the car?"

The general consensus is that the probability of winning the car is 1/3 if you don't switch and 2/3 if you do switch. But there are some implicit assumptions in this problem that cause a raging debate every time it appears on STAT-L. For example, the host may be perversely trying to goad you into a bad switch and reveals a door only when your current door has a car behind it. There are at least thirty web sites that discuss this problem. Here are three good sites:

http://www.smartpages.com/faqs/sci-math-faq/montyhall/faq.html SCI.MATH FAQ
http://www.cs.ruu.nl/wais/html/na-dir/puzzles/archive/decision.html REC.PUZZLES FAQ
http://www.ram.org/computing/monty_hall/monty_hall.html has a simulation model based on this problem.

You can also read about this problem in Engel, E. and Venetoulias, A. (1991). Monty Hall's probability puzzle. Chance, Vol 4, # 2, 6-9. and Selvin, S. (1975). A problem in probability, in "Letters to the Editor," The American Statistician, 29, 67 and 134.

The envelope exchange problem goes something like this (again from the rec.puzzles FAQ). "Someone has prepared two envelopes containing money. One contains twice as much money as the other. You have decided to pick one envelope, but then the following argument occurs to you: Suppose my chosen envelope contains $X, then the other envelope either contains $X/2 or $2X. Both cases are equally likely, so my expectation if I take the other envelope is .5 * $X/2 + .5 * $2X = $1.25X, which is higher than my current $X, so I should change my mind and take the other envelope. But then I can apply the argument all over again. Something is wrong here! Where did I go wrong? In a variant of this problem, you are allowed to peek into the envelope you chose before finally settling on it. Suppose that when you peek you see $100. Should you switch now?"

Again, there are some subtle assumptions in this problem that cause a lot of commentary. A good reference to the problem is Christensen, R. and Utts, J. (1992) "Bayesian Resolution of the 'Exchange Paradox,'" The American Statistician, 46(4), 274-276. Note also comments in the Letters to the Editor column in two separate issues the American Statistician in 1993 (pages 160, 311).

http://www.cs.ruu.nl/wais/html/na-dir/puzzles/archive/decision.html, the rec.puzzles FAQ contains a nice discussion of this problem.

The birthday problems goes something like this. There are "r" people in a room. What is the probability that two or more people have the same birthday?

Assuming uniform probabilities for each birthdate, the probability of a match is 1-(n!/(n^r)*(n-r)!) where n equals the number of days in a year and r equals the number of people in the group. For r=23, the probability exceeds 0.5. A nice summary of this problem with extensions into non-uniform birthdates is Nunnikhoven, T.S. (1992) "A Birthday Problem Solution for Nonuniform Birth Frequencies," The American Statistician, 46(4), 270-274.

http://pascal.dartmouth.edu/~zhu/applets/Birthday/Birthday.java is a Java applet for computing these probabilities.
http://www.mste.uiuc.edu/reese/birthday/intro.html has a simulation of the birthday problem.

12 What are some of the problems with stepwise regression?

14 Can someone provide me with references and/or books about [topic]?

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