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Wednesday, April 09, 2008
The Monte Hall Problem

Neatly illustrated at the New York Times. This problem drove me absolutely batty when Marilyn Vos Savant discussed it in her Parade column one day; I was convinced that she was wrong, but a simple experiment (like that shown at the Times) proves that it's indeed right.

Basically, the Monte Hall problem goes like this: Monte (of Let's Make a Deal fame) shows you three doors. Behind two of them are gag prizes, while behind the third is a very valuable prize. You pick a door. Before revealing what's behind that door, Monte now opens one of the other doors to reveal one of the gag prizes. Should you change doors, or should you stick with your original pick.

The answer, as you'll see if you try it a few times, is that you should change doors. The reason is that by revealing where one of the gag prizes was located, Monte has given you additional information that you did not have when you made your original choice.

It's pretty easy to work it out. Suppose the car is behind door number one, with the gags behind the other two. You have three choices, and odds are only 33% that you will pick the right one. But if you pick one of the wrong doors, and then switch, you are now guaranteed to choose the right one. So 67% of the time, you should switch, and 33% of the time you should stick.

John Tierney's column contains a similar problem:

1. Mr. Smith has two children, at least one of whom is a boy. What is the probability that the other is a boy?

You might think that "logically" it's 50%. But this one's a little tricky, so it helps to map out the possibilities. There are four possibilities for a parent with two children:

1. The first child was a girl and the second child was a girl.
2. The first child was a girl and the second child was a boy.
3. The first child was a boy and the second child was a girl.
4. The first child was a boy and the second child was a boy.

But we know that the first possibility is wiped altogether as at least one of the children is a boy. This means that the other three possibilities are the ones we have to account for, and it's pretty easy to see that in only one of the three the other child was a boy. Note however, that if there were more information, the odds would change. For example if we knew that the older child was a boy, then the odds that his younger sibling was a boy would be one in two, or 50%

It goes to show you that logic can sometimes lead you to the wrong conclusion.
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