The story of the three doors

Behind one of three doors is a prize; the other two hide nothing. You pick a door. Then the host, who knows where the prize is, opens one of the other doors and shows it is empty. Then he asks: “Do you want to switch?” The answer is yes. Switching raises your chance of winning from one third to two thirds. Most people think it is “fifty-fifty”, which is why the puzzle became famous.

Why switching wins

The door you first picked has a one-in-three chance of hiding the prize, and a two-in-three chance of being wrong. When the host opens a door, that chance does not change. So split the cases in two.

• You first picked the prize (one in three). Switching loses.
• You first picked an empty door (two in three). The host cannot open the prize door, so the one remaining door must hold it. Switching wins.

Switching wins exactly when your first pick was wrong, which happens two times in three. With ten doors, your first pick is wrong nine times in ten; after the host opens eight empty doors, switching wins with probability nine in ten. This site's program enumerated every combination of prize position and first pick for 3 doors and for 10 doors, and confirmed that switching wins 2/3 and 9/10 of the time.

There is a condition, though: the host knows where the prize is, always opens an empty door, and always offers the switch. If the rules differ, for example if the host does not know and just happens to open an empty door, the answer can change.

The argument it started

The name comes from the host of an American television show. The statistician Steve Selvin used it in a 1975 letter to a journal, which is regarded as the earliest published example. Similar puzzles are older: Bertrand's “three boxes” of 1889 and the “three prisoners” that Martin Gardner described in 1959 are said to have the same structure.

In September 1990, Marilyn vos Savant answered a reader's question in her Parade magazine column by saying you should switch. A flood of letters arguing she was wrong followed. By her own estimate there were about ten thousand, and some were reported to come from people with doctorates. In 1991 The New York Times put the story on its front page.

The mathematician Paul Erdős is said to have first believed it was fifty-fifty, and to have been convinced only after seeing a computer simulation, according to the recollection of his friend Andrew Vázsonyi. Even experts find it easy to get wrong.

How to think about it

• Watch what the host does. He never opens your door or the prize door. That restriction is what makes his action informative.
• Fix your first pick and split into two cases. Did you pick the prize or an empty door? One third versus two thirds shows the answer.
• Imagine 100 doors. You pick one, the host opens 98 empty ones and leaves a single door. Which is more likely to hold the prize: your first pick, or the one he went out of his way to leave closed?
• Just try it. Play a few hundred rounds and count the wins for switching and for staying. That convinces most people.

On the puzzle page you can pick doors yourself. The automatic trials run both strategies on the same rounds, so you can see the gap between them.