# [T03] The game show puzzle

As a final test of your understanding of probability, try the following rather famous puzzle.

Imagine that you are a contestant on a television game show. You are shown three large doors. Behind one of the doors is a new car, and behind each of the other two is a goat. To win the car, you simply have to choose which door it is behind. When you choose a door, the host of the show opens one of the doors you have not chosen, and shows you that there is a goat behind it. You are then given a choice; you may stick with your original choice, or you may switch to the remaining closed door.

What should you do to maximize your chances of winning the car? Think about it for a while, and when you have decided, read the two arguments below and decide which is right.

Argument 1. No need to switch: Suppose you choose door number 1. The probability that the car is behind door 1 is initially 1/3 (since there are three doors, and the car has an equal chance of being behind each). Then suppose the host opens door number 3 and shows you that there is a goat behind it. We then need to calculate a conditional probability--the probability that the car is behind door 1, given that there is a goat behind door 3. Since there are only two doors left, and there is an equal chance that the car is behind each of them, this probability is 1/2. But similarly, the probability that the car is behind door 2, given that there is a goat behind door three, is also 1/2. So whether you stick with door 1 or switch to door 2, your chance of winning is 1/2. So it really makes no difference whether you switch or not.

Argument 2. You should switch: Suppose you choose door number 1. There are three possibilities; either the car is behind door 1, or door 2, or door 3. Each of these possibilities has the same probability (1/3). In each of the three cases, consider which door the host will open. If the car is behind door 1, the host could open either door 2 or door 3. In this case, if you stick with your original choice you win the car, but if you switch to the remaining door you lose. If the car is behind door 2, the host will open door 3. In this case, if you stick with your original choice you lose, but if you switch, you win. Finally, if the car is behind door 3, the host will open door 2. Again, if you stick with your original choice you lose, but if you switch, you win. Remember that each of the three possibilities has a probability of 1/3, and note that they are mutually exclusive (the car is only behind one door). If you switch, you will win in two cases out of three (probability 2/3), but if you stick you will only win in one case out of three (probability 1/3). So you should switch doors, since it doubles your chance of winning.