Mathematics and the Game Show

Last week, I saw a reference on the Coding Horror blog about an interesting problem posed and answered on the website of Marilyn vos Savant. Marilyn is the woman who, for a while, was listed in the Guinness Book of World Records as the holder of the highest recorded IQ. I've always enjoyed her column in Parade magazine. Here is the question:

You are on a game show and they show you three doors of which you can open one. Behind one of the doors is a good prize (say a new car) and the other two doors hide things you wouldn't want (I think they used goats in the example). You pick one of the doors but they don't open it yet. Then the host, knowing which door hides the car, opens one of the other doors you didn't pick that has the goats. Then the host asks you, "Do you want to keep your door or trade it for the other door that isn't open?" The question then is, "Should you change your selection or stay with the one you originally selected?"

Most people think that the answer is that it doesn't make any difference. That's what I first thought, too. It seems there is a 50-50 chance that your original door holds the car and you can't do any better by switching. But in the 1990 Parade article, Marilyn said you should change your selection. As a matter of fact, she says that you have twice as much chance of winning if you change which door you want to open!

Her original explanation of why this is true didn't convince a lot of people - including a number of math professors in college. She got a lot of nasty mail saying that she was wrong and that she was afraid to admit it. Then she did what everyone should have done from the beginning. She wrote a simple table listing every possible outcome and just counted up the good and the bad outcomes to show that it is indeed better to change your selection once the host has opened one of the doors that doesn't hold the car. There are good lessons for all of us here: 1) Math doesn't have to be hard, 2) Even when we think we know the answer, it's best to write it down, 3) We shouldn't be so sure of ourselves until we've checked the facts. I suggest you go to Marilyn's discussion of this problem and to see her table to convince yourself.

Of course, the whole premise of this is based on the idea that you would want a car more than you would want the goats :-) Perhaps you'd rather have the goats. Then I guess it's better to stay with your original selection. By the way, the picture of the goats at the top of this article is from http://www.keeping-goats.com/