contestant originally chose 1 and the host revealed that the car is net behind 2, what is the conditional probability that the car is behind door 3? With the same given conditions1 what is the conditional probability the the car is behind door 1? The answer to these questions depends on the assumptions. Assume (based on many observations of this game) that initially the car was equally likely to be behind any of the 4 doors, and if the contestant's original choice was the door with the car, then with probability 2/3, the host Chooses to open one of the doors that was not chosen? with equal probability for any of the 3 choices. Also assume that if the contestant's original choice was not the door with the car, then with probability 1 #1, the host chooses to open one of the doors that was not chosen and does not contain the car, with equal probability for any of the 2 choices. Suppose a contestant on a game show is given the choice of four closed doors and will win the prize behind the chosen door. Behind one of the doors is a car, behind the others, goats. The contestant chooses door 1. The host, who knows what's behind the doors opens door 2, which has a goat. He then offers to let the contestant choose between doors 3 or 4 instead of 1, if the contestant would like to do so. Would the contestant improve the chances of winning the car by choosing between doors 3 and 4 instead of 1? Given that the