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Goat and Car Paradox *January 29, 2009*

*Posted by keithkchan in fun stuffs.*

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When my roommate watched the movie 21, he was curious about the Goat and Car Paradox posed in the movie. In the movie an MIT student Ben was challenged the following problem by the professor in the lecture theatre. Here is the problem:

Suppose you are in a TV show. There are 3 doors. Behind one of them there is a car, and behind the other two there is a goat. You are allowed to choose a door, you will get whatever behind the door. Suppose you choose Door no. 1. The host of the TV show knows which one is the car. After you choose the first door, he opens a door with a goat among the door 2 and 3, say 3. Now the host asks if you want to change your option to Door no. 2 or remains the same as your initial decision.

It seems that there are 2 doors remaining, the chance you get it right is 50-50, why should I bother to change? In the movie 21, the guy Ben chooses to change to Door no. 2. He give some explanations. But in my roommate and my opinion, his explanation can only make the problem more mysterious. I remembered I read about this paradox many years ago, but I forget why it is better to change. So we get the pen and paper and start the discussion.

To make things clear, you should draw the tree diagram. Suppose you choose the car door initially (with probability 1/3), you change you get a goat with probability 1/3. Suppose you choose a goat. Here is the point. The host can’t open the car door, so he must open the remaining goat door. Then you are sure that the goat is in the remaining door. If you change, you get a car with probability 2/3. The thing that breaks the symmetry is that the host does not open the door randomly, he has to deliberately open the goat door. In this act he gives you more information. This example shows that conditional probability is sometimes tricky.

Here is a variant. Suppose the host does not know which door contains the car. He opens one of the other door randomly. If he opens the door with a car, you lose. Now, the door has been opened with a goat inside. Do you want to change? The answer is in the comment. BTW, you may want to watch the movie 21 if you have time.

First draw the tree diagram as in the post. Since you know he does not open the one with a car, you cross out the possibilities that he open the car door. From the remaining diagram, you can immediately see that the probability is 50-50, so it doesn’t matter if you change or not.