The three door quiz show puzzle shows how wrong intuition can be when dealing with probabilities. A contestant in a game show may win a car by choosing which of three doors the car is behind.
When you make your guess, the game show host opens one of the two doors you did not choose, and there is no car behind it. The host now gives you the opportunity to change your mind and choose the other unopened door.
Do you change your guess or keep your original choice?
Many people assume that the car is equally likely to be behind either door and many stick to the door they have chosen. This ignores the fact that there is a symmetry broken by the host choosing from the other two doors which one to open.
When you pick your door, before any door is opened, the probability that you picked correctly is 1/3 and the probability that you chose incorrectly is 2/3.
Now the host opens one of the other doors. He did not choose the door you chose, just because you chose it. If he had picked from all three doors which door to open to reveal no car, then the car could equally likely have been behind the two remaining doors.
Suppose the three doors are A, B and C, and suppose the contestant chose A, then the host will choose B or C will equal probability.
The probability the contestant chose correctly is
So whichever other door is opened, the probability that the remaining door hides the car is