The Monty Hall problem
Three doors, one car. The host opens a goat door. Should you switch?
Steve Selvin, 1975; Marilyn vos Savant's column, 1990
The game show that fooled the experts
On a TV game show, there are three doors. Behind one is a brand-new car; behind the other two are goats. You pick door 1. The host, who knows where the car is, opens door 3 and shows you a goat. Then he smiles and asks: "Do you want to switch to door 2?"
Almost everyone says it doesn't matter. Two doors left, one car: surely it's 50–50.
In 1990 a writer named Marilyn vos Savant answered this question in a magazine column. Her answer: always switch, because switching wins two times out of three. She received about ten thousand letters, many from people with maths doctorates, telling her she was wrong.
She wasn't. Even the great mathematician Paul Erdős reportedly refused to believe it until he was shown a computer simulation.
The puzzle
After the host opens a goat door, switching wins twice as often as sticking, even though there are only two doors left.
See it for yourself
What's really going on
When you first pick, you have a 1 in 3 chance of choosing the car, and a 2 in 3 chance that the car is behind one of the other two doors. The host's action doesn't change that: he will always open a goat door, whatever you picked. If your first pick was wrong (2 times in 3), the car must be behind the remaining door, so switching wins.
Another way to see it: imagine 100 doors. You pick one, and the host opens 98 goat doors, leaving yours and one other. Would you stick with your 1-in-100 guess? The host's knowledge is what makes the doors unequal.
Think about it
What if the host didn't know where the car was, and just happened to open a goat door? Would switching still help?