Let's Make a Deal ProbabilityDate: 06/01/99 at 11:15:53 From: Brion McGinn Subject: Let's Make a Deal Probability In the TV show "Let's Make a Deal" they show you 3 doors, of which you pick one. Then Monty shows you what's behind one of the doors you did not pick, and the door he shows you is a loser. Then he asks you if you would like to change doors. From a probability standpoint, should you change doors? In other words, does either the act of his showing you a door or his knowing the answer when he shows you a door affect the probabilty? If there is no effect of his knowing or showing, you start out choosing with a 33% chance of being right. Then when the door is shown, your probability changes from 33% to 50%, and there is no benefit to switching or not switching. However, if his knowing has an influence, then let's look at what happens if you choose door number 1. If door 1 is right (1/3 of time): Monty shows you door 2 (1/6 of time) you should NOT switch Monty shows you door 3 (1/6 of time) you should NOT switch If door 2 is right (1/3 of time): Monty shows you door 3 (1/3 of time) you should switch If door 3 is right (1/3 of time): Monty shows you door 2 (1/3 of time) you should switch Following this, assume Monty shows you door 2... then 1/3 of the total times you should switch, and 1/6 of the total times you should not switch. Or, you are twice as likely to be right by switching than by not switching. Which of the two circumstances is the case? Is it, or is it not beneficial to switch? Date: 06/01/99 at 11:45:08 From: Doctor Mitteldorf Subject: Re: Let's Make a Deal Probability Dear Brion, You're right that it depends on what Monty knows and what he intends. Take a look at our FAQ answer at http://mathforum.org/dr.math/faq/faq.monty.hall.html Here's another paradox, from my friend Uri Wilensky, that you might enjoy: You're shown two envelopes with money inside, and you are told that one envelope has twice as much money as the other. You can pick A or B. You pick A, and open it, to find $100. Now you're given the opportunity to switch. If you switch, you have a 50% chance of losing $50 and a 50% chance of gaining $100. So, it would seem that switching is the thing to do. But, wait a minute - you have no new information that you didn't have before. Can it really be that the "second envelope" usually gives you a better prize than the first? - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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