The Monty Hall problem is a brain teaser, in the form of a probability puzzle, based nominally on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from reader Craig F. Whitaker's letter quoted in (and solved by) Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:
Savant's response was that the contestant should switch to the other door. By the standard assumptions, the switching strategy has a probability of winning the car, while the strategy of keeping the initial choice has only a probability.
When the player first makes their choice, there is a chance that the car is behind one of the doors not chosen. This probability does not change after the host reveals a goat behind one of the unchosen doors. When the host provides information about the two unchosen doors (revealing that one of them does not have the car behind it), the chance of the car being behind one of the unchosen doors rests on the unchosen and unrevealed door, as opposed to the chance of the car being behind the door the contestant chose initially.
The given probabilities depend on specific assumptions about how the host and contestant choose their doors. An important insight is that, with these standard conditions, there is more information about doors 2 and 3 than was available at the beginning of the game when door 1 was chosen by the player: the host's action adds value to the door not eliminated, but not to the one chosen by the contestant originally. Another insight is that switching doors is a different action from choosing between the two remaining doors at random, as the former action uses the previous information and the latter does not. Other possible behaviors of the host than the one described can reveal different additional information, or none at all, leading to different probabilities. In her response, Savant states:
Many readers of Savant's column refused to believe switching is beneficial and rejected her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling Savant wrong. Even when given explanations, simulations, and formal mathematical proofs, many people still did not accept that switching is the best strategy. Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating Savant's predicted result.
The problem is a paradox of the veridical type, because the solution is so counterintuitive it can seem absurd but is nevertheless demonstrably true. The Monty Hall problem is mathematically related closely to the earlier three prisoners problem and to the much older Bertrand's box paradox.
Paradox Steve Selvin wrote a letter to the American Statistician in 1975, describing a problem based on the game show Let's Make a Deal, dubbing it the "Monty Hall problem" in a subsequent letter. The problem is equivalent mathematically to the three prisoners problem described in Martin Gardner's "Mathematical Games" column in Scientific American in 1959 and the Three Shells Problem described in Gardner's book Aha Gotcha.
Standard assumptions Ambiguities in the Parade version do not explicitly define the protocol of the host. However, Marilyn vos Savant's solution printed alongside Whitaker's question implies, and both Selvin and Savant explicitly define, the role of the host as follows: The host must always open a door that was not selected by the contestant. The host must always open a door to reveal a goat and never the car. The host must always offer the chance to switch between the door chosen originally and the closed door remaining. It is also presumed that the car is placed (uniformly) at random behind one of the doors and that, if the player initially chooses the car, then the host's choice of which goat-hiding door to open is also made (uniformly) at random.
Under these standard assumptions, the probability of winning the car after switching is . Varying the assumptions can change the probability of winning by switching doors as detailed in the section below.
These assumptions specify the mathematical model of host behavior and prize distribution being analyzed. In a decision-theoretic reading, whether switching is rational for the contestant also depends on the contestant knowing, or having reason to assume, that the host follows this protocol.
Simple solutions
The solution presented by Savant in Parade shows the three possible arrangements of one car and two goats behind three doors and the result of staying or switching after initially picking door 1 in each case:
Behind door 1 Behind door 2 Behind door 3 Result if staying at door #1 Result if switching to the door offered Goat Goat Car Goat Car Goat Car Goat Goat
A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three.
An intuitive explanation is that, if the contestant initially picks a goat (2 of 3 doors), the contestant win the car by switching because the other goat can no longer be picked the host had to reveal its location whereas if the contestant initially picks the car (1 of 3 doors), the contestant win the car by switching. concluding remarks Using the switching strategy, winning or losing thus only depends on whether the contestant has initially chosen a goat ( probability) or the car ( probability). The fact that the host subsequently reveals a goat in one of the unchosen doors changes nothing about the initial probability.
Most people conclude that switching does not matter, because there would be a 50% chance of finding the car behind either of the two unopened doors. This would be true if the host selected a door to open at random, but this is not the case. The host-opened door depends on the player's initial choice, so the assumption of independence does not hold. Before the host opens a door, there is a probability that the car is behind each door. If the car is behind door 1, the host can open either door 2 or door 3, so the probability that the car is behind door 1 the host opens door 3 is If the car is behind door 2 – with the player having picked door 1 – the host open door 3, such the probability that the car is behind door 2 the host opens door 3 is These are the only cases where the host opens door 3, so if the player has picked door 1 and the host opens door 3, the car is twice as likely to be behind door 2 as door 1. The key is that if the car is behind door 2 the host open door 3, but if the car is behind door 1 the host can open either door.
Another way to understand the solution is to consider together the two doors initially unchosen by the player. As Cecil Adams puts it, "Monty is saying in effect: you can keep your one door or you can have the other two doors". The chance of finding the car has not been changed by the opening of one of these doors because Monty, knowing the location of the car, is certain to reveal a goat. The player's choice after the host opens a door is no different from if the host offered the player the option to switch from the original chosen door to the set of remaining doors. The switch in this case clearly gives the player a probability of choosing the car.
As Keith Devlin says, "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is . I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.
Savant suggests that the solution will be more intuitive with 1,000,000 doors rather than 3. In this case, there are 999,999 doors with goats behind them and one door with a prize. After the player picks a door, the host opens 999,998 of the remaining doors. On average, in 999,999 times out of 1,000,000, the remaining door will contain the prize. Intuitively, the player should ask how likely it is that, given a million doors, they managed to pick the right one initially. Stibel et al. proposed that working memory demand is taxed during the Monty Hall problem and that this forces people to "collapse" their choices into two equally probable options. They report that when the number of options is increased to more than 7 people tend to switch more often; however, most contestants still incorrectly judge the probability of success to be 50%.
Savant and the media furor Savant wrote in her first column on the Monty Hall problem that the player should switch. She received thousands of letters from her readersthe vast majority of which, including many from readers with doctorate degrees, disagreed with her answer. During 1990–1991, three more of her columns in Parade were devoted to the paradox. Numerous examples of letters from readers of Savant's columns are presented and discussed in The Monty Hall Dilemma: A Cognitive Illusion Par Excellence.
The discussion was replayed in other venues (e.g., in Cecil Adams's The Straight Dope newspaper column) and reported in major newspapers such as The New York Times.
In an attempt to clarify her answer, she proposed a shell game to illustrate: "You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are , agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger." She also proposed a similar simulation with three playing cards.
Savant commented that, though some confusion was caused by readers' not realizing they were supposed to assume that the host must always reveal a goat, almost all her numerous correspondents had correctly understood the problem assumptions, and were still initially convinced that Savant's answer ("switch") was wrong.
Confusion and criticism Sources of confusion
When first presented with the Monty Hall problem, an overwhelming majority of people assume that each door has an equal probability and conclude that switching does not matter. Out of 228 subjects in one study, only 13% chose to switch. In vos Savant's book The Power of Logical Thinking, cognitive psychologist Massimo Piattelli-Palmarini writes: "No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they on it, and they are ready to berate in print those who propose the right answer". Pigeons repeatedly exposed to the problem rapidly learn to always switch, unlike humans.
Most statements of the problem, notably the one in Parade, do not match the rules of the actual game show and do not fully specify the host's behavior or that the car's location is randomly selected. However, Krauss and Wang argue that people make the standard assumptions even if they are not explicitly stated.
Although these issues are mathematically significant, even when controlling for these factors, nearly all people still think each of the two unopened doors has an equal probability and conclude that switching does not matter. This "equal probability" assumption is a deeply rooted intuition. People strongly tend to think probability is evenly distributed across as many unknowns as are present, whether or not that is true in the particular situation under consideration.