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### Let's Make a Deal Probability

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Date: 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?
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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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Associated Topics:
High School Probability
High School Puzzles

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