About 10,000 readers told a columnist her maths was wrong. She was right.
Three doors, one car, two goats. You pick a door; the host, who knows where the car is, opens another to reveal a goat. Should you switch? Switching wins two times in three. When a magazine columnist said so in 1990, around a thousand PhDs wrote in to insist she'd blundered.
The puzzle is named after Monty Hall, host of the American game show Let's Make a Deal, though the real show never offered contestants this switch. The statistician Steve Selvin posed it in letters to The American Statistician in 1975. It became notorious in September 1990, when Marilyn vos Savant answered a reader's question about it in her Parade magazine column and said the contestant should switch.
The key is that the host's choice is not random. He must open an unchosen door, must reveal a goat, and must offer the switch. Your first pick has a one-in-three chance of being the car, so two times in three the car is behind one of the other doors. When the host opens the goat door among those two, he effectively tells you which of them to take. Switching wins whenever your first guess was wrong, which is two-thirds of the time. Vos Savant's intuition pump: imagine a million doors, and the host opens all but one of the ones you didn't choose.
The reaction was ferocious. By her estimate, some 10,000 people wrote to the magazine, nearly 1,000 of them with PhDs, most saying she was wrong. Even the great mathematician Paul Erdős reportedly stayed unconvinced until he saw a computer simulation. In one study, only 13 per cent of 228 people chose to switch. Pigeons, repeatedly exposed to the game, quickly learn to always switch; humans are slower.
The problem has older relatives, including Joseph Bertrand's box paradox of 1889 and a 'three prisoners' puzzle Martin Gardner published in 1959. Its lasting lesson is that probability depends on how information is revealed, not just on what you can see. Change the host's rules, for example by letting him open doors at random, and the answer changes too.
Source: Wikipedia — Monty Hall problem · Text summarised from Wikipedia (CC BY-SA 4.0)
