Michael Shermer posed the following question in this month's SciAm:
Imagine that you are a contestant on the classic tv game show "Let's Make a Deal." Behind one of the three doors is a brand-new car, behind the other two are goats. You choose door #1. Monty opens door #2, revealing a goat, and asks if you want to stay with door #1 or change your answer to door #3. Do you improve your odds by switching?
Dr. Shermer argues that you do. To quote him, "Here is why. There are three possible three-doors configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good. In (1) you lose by switching, but in (2) and (3) you can win by switching."
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Dr. Shermer is much smarter than me, so I'll give him the benefit of the doubt, but I don't get it. I don't think the fact that it started as a 3-door configuration is relevant. In the original 3-door configuration, by random change you have a 33ish% chance of picking the car; in the current 2-door configuration you have a 50% chance of picking the car. Schroedinger might agree with Shermer, because Monty Hall obviously knows where the car is-but to me that seems to confuse superpositioning with discrete probability trials. The way I remember statistics, the odds of flipping a coin to "heads" 30 times in a row is astronomically low-but if you have already flipped a coin to "heads" 29 times in a row, the odds that you will flip the same coin to "heads" on the next flip remains at 50%. Shermer and I agree on that much, at least-later, in the same article, he talks about the fallacy of "hot streaks."
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Now I hate mathematics, but when I saw this posted in the journal of another friend I was immediately reminded of the movie 21 (I know I know, why must everything seem to relate to movies???)
In the movie they explained that by taking out one third of the choices your chance of winning if you switched was raised to some 66% or there abouts, increasing the probability by 33 percent or something. *shrug*
My problem with this bit here:
"Dr. Shermer argues that you do. To quote him, "Here is why. There are three possible three-doors configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good. In (1) you lose by switching, but in (2) and (3) you can win by switching." "
is that by having chosen door number one and having the goat revealed behind door number two, You only have Option One and Option Three left. There is no option 2 where switching is beneficial to your cause anymore (unless as some have stated, your 'cause' is to get the goat) So his reasoning for why switching is a good idea, is no longer valid.
wiki ( http://en.wikipedia.org/wiki/Monty_Hall_problem ) has a good article explaining it but i think a few of their visual aids are mistaken... lending that some of the situations on where the 'prizes' are placed have higher probabilities than they really do.
