Most people come to the conclusion that switching does not matter because there are two unopened doors and one car and that it is a 50/50 choice. This would be true if the host opens a door randomly, but that is not the case; the door opened depends on the player's initial choice, so the assumption of independence does not hold. Before the host opens a door there is a probability that the car is behind each door. If the car is behind door 1 the host can open either door 2 or door 3, so the probability that the car is behind door 1 and the host opens door 3 is × = . If the car is behind door 2 (and the player has picked door 1) the host must open door 3, so the probability that the car is behind door 2 and the host opens door 3 is × 1 = . These are the only cases where the host opens door 3, so if the player has picked door 1 and the host opens door 3, the car is twice as likely to be behind door 2 as door 1. The key is that if the car is behind door 2 the host must open door 3, but if the car is behind door 1 the host can open either door.
