Monty Hall problem

The standard formulation involves three doors: one hides a car, the other two goats. The contestant picks a door; the host, who knows what is behind each door, opens another door revealing a goat. The contestant is then offered the chance to switch to the remaining door. The counterintuitive solution is that switching doubles the chance of winning from 1/3 to 2/3. This is because the host's action provides information: the probability that the car is behind the initially chosen door remains 1/3, while the probability that it is behind the other unopened door becomes 2/3.² The reasoning is often clarified using Bayes' theorem or by enumerating all possible outcomes. The problem assumes the host always opens a door with a goat and always offers the switch.

The problem first appeared in 1975 in a letter by Steve Selvin to The American Statistician.³ It gained widespread attention in 1990 when Marilyn vos Savant, listed in the Guinness Book of World Records for highest IQ, addressed it in her Parade magazine column. She correctly stated that switching is advantageous, but her answer provoked thousands of angry letters, including from mathematicians and PhDs, many of whom insisted the odds were 50–50. The ensuing controversy brought the problem to the public eye and stimulated academic discussion. Monty Hall himself later commented on the problem, noting that the host's behavior in the actual show sometimes differed from the puzzle's strict assumptions.⁴

Numerous variants explore the sensitivity of the solution to the host's behavior. For example, if the host randomly picks a door to open and it happens to reveal a goat, the probability of winning by switching drops to 1/2. Conversely, with n doors and the host opening all but one goat door, the switch advantage becomes (n-1)/n.⁵ Other modifications include a biased host, multiple stages, or the "Monty Hall problem" applied to quantum mechanics and decision theory. The problem has been used to teach conditional probability, Bayesian inference, and the pitfalls of intuitive reasoning. In cognitive science, it serves as a benchmark for studying belief revision and the difficulty of updating probabilities.¹

Despite its fame, few know that the problem was originally posed in a different form: Selvin's 1975 letter described a "box problem" with three boxes, one containing a prize. The game show host was not named until later, but the puzzle became forever linked to Monty Hall. Hall himself noted that in the actual show, he sometimes offered cash to keep the contestant from switching, muddying the pure mathematical scenario.⁴ Another surprising fact: a 2011 study found that even after being shown the correct answer, many people still refuse to switch, exhibiting a persistent cognitive bias.⁶ The problem has also been used in legal and medical contexts to illustrate how diagnostic tests can mislead when base rates are ignored.

Conditional probability

The probability of an event given that another event has occurred.

Bayes' theorem

A mathematical formula for updating probabilities based on new evidence.

Host's knowledge

The crucial assumption that the host knows the location of the car and always reveals a goat.