Science · Mathematics

The Monty Hall problem

1975 / 1990

Mathematics — probability

A probability puzzle in which a contestant picks one of three doors, behind one of which is a car and behind the others goats. The host, who knows what is where, opens a different door to reveal a goat and offers a switch. Switching wins two times in three; staying wins one in three. Named after the host of the American game show Let's Make a Deal, it was posed in that form by Steve Selvin in 1975 and became notorious in 1990 when Marilyn vos Savant gave the correct answer in her Parade column and was contradicted by thousands of readers, including many with doctorates.

How to spot them

▸Three doors, a car and two goats, and a host who knows where the car is
▸Switching wins two thirds of the time; the intuition that says 50-50 is wrong
▸Named after the host of the American game show Let's Make a Deal
▸Marilyn vos Savant's 1990 Parade column drew about 10,000 letters telling her she was wrong
▸Equivalent to Bertrand's box paradox and the three prisoners problem

Key facts

Three doors, two goats and a car, with the host opening a door you did not pick

The set-up, drawn from the American game show Let's Make a Deal

A 1975 letter to The American Statistician that first posed it in this form (1975)

Steve Selvin of the University of California, Berkeley

A September 1990 newspaper column that drew some 10,000 letters of correction, nearly 1,000 from PhDs (1990)

Marilyn vos Savant's 'Ask Marilyn' column in Parade magazine

Paul Erdős's refusal to accept the answer until shown a computer simulation

Recounted in Paul Hoffman's The Man Who Loved Only Numbers

The counter-intuitive advice to abandon your first choice once a door is opened

Switching wins 2/3 of the time; staying wins 1/3

A 2008 scene in the film 21, in which an MIT student explains the answer to a professor (2008)

The film about card-counting students, based on Ben Mezrich's Bringing Down the House

The 1959 three prisoners puzzle of Martin Gardner, mathematically the same thing (1959)

Gardner's Scientific American column; Bertrand's box paradox of 1889 is another version

Key ideas

Your first choice is right one time in three. If you were right, switching loses; if you were wrong — two times in three — the host is forced to leave the car behind the remaining door, so switching wins.
The host's knowledge is essential. If he opened a door at random and happened to reveal a goat, the odds really would be even. This is why the puzzle is a lesson in conditional probability, not arithmetic.
Scaling up makes it obvious: with a hundred doors, you pick one, the host opens ninety-eight goats, and almost nobody would decline to switch.

Quick-fire facts

•Monty Hall (1921–2017), born Monte Halparin in Winnipeg, hosted Let's Make a Deal from 1963
•Steve Selvin posed the problem in a letter to The American Statistician in 1975 and gave it the host's name
•Marilyn vos Savant answered it in Parade on 9 September 1990; roughly 10,000 readers wrote in, most insisting she was wrong
•About 1,000 of the objectors held doctorates, many of them mathematicians
•Switching wins with probability 2/3, staying with probability 1/3
•Paul Erdős reportedly remained unconvinced until Andrew Vázsonyi showed him a computer simulation
•The puzzle is equivalent to Martin Gardner's three prisoners problem of 1959 and to Bertrand's box paradox of 1889
•Monty Hall himself pointed out that on the real show he was not obliged to offer a switch, which changes the analysis entirely
•The problem features in the 2008 film 21 and in Mark Haddon's novel The Curious Incident of the Dog in the Night-Time
•It is a standard illustration of Bayes' theorem and of the difference between conditional and unconditional probability

Drill it — 9 cards

1 / 9new card

Sources

Wolfram MathWorld — Monty Hall Problem
Paul Hoffman — The Man Who Loved Only Numbers