You pick a door. The host, who knows where the car is, opens one of the other two and shows a goat. Now you may stay with your door or switch to the last one.

In 1990 the columnist Marilyn vos Savant, listed by Guinness for the highest recorded IQ (a category it later dropped), answered: switch. By her own estimate about 10,000 letters came in, most telling her she was wrong, and close to 1,000 were signed by PhDs. In 1995 Paul Erdős, one of the great mathematicians of the century, heard the puzzle at a friend's home and said it should make no difference. A decision tree did not move him, so the friend, Andrew Vazsonyi, let a computer play the game 100,000 times. Staying won about one game in three, switching two in three. Erdős, Vazsonyi wrote, "still did not understand the reason why, but was reluctantly convinced".

The reason: your first pick is right one time in three. So two times in three the car is behind one of the other doors, and the host has just shown you which of them it is not. This holds because the host knows where the car is and always opens a goat door; if he opened one by accident, it would be fifty-fifty.

The games on screen are our own run of 100,000.

Did you switch?

Sources: Marilyn vos Savant, "Game Show Problem" (her Parade columns, 1990-91); John Tierney, "Behind Monty Hall's Doors", New York Times, 21 July 1991; Andrew Vazsonyi, "Which Door Has the Cadillac?", Decision Line 30(1), 1999; Jeffrey Rosenthal, "Monty Hall, Monty Fall, Monty Crawl", Math Horizons, 2008.
Sounds: Freesound, CC0 (Brighella, satoristudios3); our own, made in code (earshot, CC0).