The Monty Hall Problem

I heard of the Monty Hall Problem years ago, and believed that Marilyn vos Savant was full of crap. Today my friend Adam Dingle explained to me why she wasn't. It took me a long time to understand, and even longer to believe. But here's my explanation.

You have a 1/3 chance of picking the right door. Therefore you have a 2/3 chance of picking the wrong one. If you've guessed wrong, the host eliminates all but the winning door. This is true because he's promised that after you choose, he'll narrow it down to two doors, one of which contains the winner.

Thus, you now have two choices: stay with your original choice, which you knew had a 1/3 chance of being right; or switch to the other door, which must be the winning door if you earlier picked wrong.

Marilyn's million-door example is excellent, but she chose a rhetorical rather than clear explanation for it.

You have a one-in-a-million chance of picking the right door on the first guess. Then the host eliminates all but your pick and another door, one of which is the winner. If you stay, then you believe that you picked the winner on the first guess. If you switch, then you believe that you picked wrong, and are now switching to the winning one. It's much more likely you guessed wrong, forcing the host to reveal the winning door (since he had to make sure that the remaining two doors contained the winner).

The devilish part of this problem is that there are three doors, not a million. This means that the host eliminates only one door of the remaining two, so you don't realize that he's actually eliminating every remaining door that's a loser.