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OFF TOPIC -- A question of logic

Posted By: Timothy Chow
Date: Saturday, 2 May 2015, at 6:17 p.m.

In Response To: OFF TOPIC -- A question of logic (Mike Clapsadle)

The solution may be found here, and there's even a whole book on the subject. Note that there are several tacit assumptions being made (the show master knows what's behind the doors, the show master always opens a door, the opened door never has a prize behind it, the show master never opens the door you've chosen). However, most people still get the answer wrong even after these sources of ambiguity are removed.

The way I like to think of it is as follows. Imagine playing the game over and over again with an imaginary friend by your side. You and your friend together pick some door at random, but you both choose the same door. After the host opens a door, you always switch, and your friend never switches. Now observe:

1. You and your friend have both doors covered, so one and only one of you wins each time.

2. Your friend will win 1/3 of the time, because he will win if and only if the originally chosen door was correct. From your friend's point of view the host's opened door is irrelevant because it never influences his behavior.

It follows that you win 2/3 of the time because you win if and only if you friend doesn't win.

Now get rid of the imaginary friend.

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