OpenCourseWare on critical thinking, logic, and creativity

Try again!

This argument is incorrect. But don't feel bad--you're in good company. This argument has fooled a lot of people, including eminent mathematicians, and the reason it is incorrect is very subtle.

For comparison, consider a variant on this game. Suppose you don't tell the host which door you have chosen. The host then opens a door with a goat behind it (say, door 3), and gives you a chance to change your choice. For this game, Argument 1 is correct; given that you know there is a goat behind door 3, the probability that there is a car behind door 1 is 1/2, and the same for door 2.

Now let's go back to the original game. The difference here is that the host is not allowed to open the door you have chosen, and this means that the host often has no choice about which door to open. If you happen to have chosen the door with the car (which has probability 1/3), the host may open either of the other doors. But in the more likely event that you have chosen a door with a goat behind it (probability 2/3), the host has no choice; he has to open the other door with a goat. In this case, the door the host chooses tells you where the car is. You can use this information; it tells you that 2/3 of the time, switching doors is the right strategy.

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