In the Monty Hall problem, switching doors after the host reveals a goat makes no difference; you still have a 50/50 chance of winning.

In September 1990, Marilyn vos Savant published an answer to a reader's question in her Parade magazine column. The question described a game show scenario: a contestant picks one of three doors, the host (who knows where the prize is) opens one of the remaining doors to reveal a goat, and the contestant decides whether to stay with the original choice or switch to the other remaining door. Vos Savant replied that switching wins two-thirds of the time, not half.

Within weeks, Parade received around 10,000 letters. The majority told her she was wrong. Among them were holders of doctorates in mathematics, statistics, and related fields. A professor at the U.S. Naval Academy stated flatly that she had made an error. A statistician at George Mason University wrote that the problem was a trick question and that she had given the wrong answer. Several academics expressed concern that her error would cause lasting confusion among readers.

The correct answer is that switching wins two-thirds of the time. Considering all three possible initial choices makes this clear. If the contestant originally picks the door concealing the car, switching loses. If the contestant originally picks either of the two goat doors, the host is forced to open the remaining goat door, and switching wins. Two of the three equally likely starting positions end in a win for the switcher. The host's constraint, that he must open a losing door other than the one the contestant chose, is the key: his action is not random and carries information about where the prize is.

The confusion arose partly because the problem resembles a different problem where the 50-50 answer is correct. If a host randomly opens a door and might accidentally reveal the car, the remaining two doors do carry equal probability. The Monty Hall version, where the host always opens a losing door, is a distinct problem, and the two were sometimes conflated in probability textbooks of the 1970s and 1980s. Some textbook versions of door-switching problems did not specify whether the host knew where the prize was, producing problems with genuinely different correct answers depending on the interpretation.

The puzzle had appeared in academic literature before vos Savant's column. Steve Selvin published the problem in the American Statistician in 1975, where it attracted some letters of disagreement but relatively little notice. The 1990 column reached a mass audience and produced a public controversy that academic journals had not generated. Computer simulations written by doubting readers and run thousands of times consistently confirmed that switching won two-thirds of the time.

The name refers to Monty Hall, host of the game show 'Let's Make a Deal,' which ran from 1963 to 1977 and in later versions. The specific mechanics of the show did not always match the formalized problem, since Hall did not follow a strict protocol about when and whether to offer a switch. The classroom version is an abstraction in which the host always reveals a goat and always offers a switch, regardless of the contestant's original choice.

By 1991, cognitive psychologist Massimo Piattelli-Palmarini had described the problem as a demonstration of how reliably human intuition fails at conditional probability. Probability textbooks published after the controversy began correcting their treatment of door-switching problems, specifying the host's knowledge and intent and providing the complete probability tree. The problem became a standard component of probability and decision theory courses, presented alongside behavioral data showing that most people choose to stay rather than switch even after the correct answer is explained.