The Monty Hall Problem and Why Switching Wins

Switching doors after the host reveals a goat wins the car two thirds of the time, not one half, because the host's reveal is not random.

Switching doors after the host reveals a goat wins the car with probability two thirds, while staying with the original choice wins with probability one third. The result feels wrong to most people on first exposure because it seems like the reveal should make the two remaining doors equally likely, and the entire content of the problem is explaining exactly why it does not.

There are three doors. Behind one is a car, and behind the other two are goats, with the arrangement fixed uniformly at random before the game starts. A contestant picks one door, which is not opened. The host, who knows what is behind every door, then opens one of the two remaining doors, always revealing a goat and never the door the contestant picked. The contestant is then offered the choice to switch to the other unopened door or stay with the original pick.

The result depends critically on the host's behavior being fixed and known: the host always opens a door, always reveals a goat, and never opens the contestant's door. Versions of the puzzle that relax these conditions, for instance a host who opens a random door and sometimes reveals the car by accident, produce different numbers, which is why interviewers press on the exact rules before accepting an answer.

Before any door is opened, the contestant's first pick has a one third chance of being the car and a two thirds chance of being a goat, since the car's location is uniform across three doors. This one fact does all of the work in the problem, because the host's subsequent action never changes where the car actually is.

If the contestant's initial pick was the car, an event with probability one third, then switching loses, since both remaining doors have goats and the host simply reveals one of them. If the contestant's initial pick was a goat, an event with probability two thirds, then the host is forced to reveal the other goat, since the host cannot open the contestant's door or the car's door, which means the single remaining unopened door must contain the car. Switching in this case wins with certainty.