Monty Hall problem

Monty Hall problem, probability puzzle named for Monty Hall, the host of the American television game show Let’s Make a Deal (1963–2003). It was first posed in print in the mid-1970s and became popular in the 1990s. The puzzle is now widely cited in mathematics, decision theory, and cognitive psychology.

Premise of the puzzle

In Let’s Make a Deal, a contestant is shown three closed doors and informed by the host that behind one of the doors is a car (the prize) whereas the other two hide a goat each. The contestant is then asked to select a door. Once that is done, the host opens one of the two other doors, revealing a goat. The host then asks the contestant whether to continue with the initial choice or switch to the other closed door.

Although contestants commonly stay with their initial choice, switching doors is associated with a higher chance of winning. In fact, switching increases the chances of winning to 2/3, whereas staying with the original choice leaves the contestant with just a 1/3 chance. This has been verified both mathematically and through repeated simulations.

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Logic of switching

Switching doors makes sense when one considers the additional information provided by the host’s action. The host always opens a door that conceals a goat, never the car. By opening a particular door, the host not only reveals that it does not conceal the car but also increases the likelihood, from the contestant’s perspective, that the car is behind the other unopened door—otherwise the host would have opened that door. At first, the two doors the contestant did not choose have a combined chance of 2/3 of hiding a car behind one of them. When opening one of these two doors reveals the goat, the 2/3 chance still remains with this group, now with only one unopened door. Hence, in a manner, the chance is now transferred to that one remaining door. Sticking with the initial choice, however, retains the odds of finding the car behind the initially selected door at 1/3. Thus, over many repetitions of the game, switching wins roughly twice as often as sticking.

The following table breaks down the possible outcomes of the two scenarios: sticking or switching. Assuming that the contestant selects door 1 at the start of the game, sticking with the initial choice only leads to a win in one case. Switching, on the other hand, leads to a win in two out of three cases.

It is easier to understand the change in probability when the number of choices is increased substantially beyond 3. If, for instance, there are 100 doors and the host reveals 98 of the 99 doors that were not picked by the contestant, then the chance or probability of the car being hidden behind the remaining nonselected closed door would be 99/100. This would be much greater than the probability of the car being concealed by the door initially selected by the contestant, which is 1/100. Most people are likely to switch options in this case as the asymmetry in probabilities becomes more obvious and therefore convincing.

Mathematics behind the puzzle

Bayes’s theorem, also called conditional probability, provides the formal mathematical framework for updating the probability of winning after receiving new information from the host. The theorem, which calculates the probability (P) of a hypothesis (h) given evidence (e), states:$P \left(hypothesis\mid evidence\right)=\frac{P \left(evidence\mid hypothesis\right)\times P \left(hypothesis\right)}{P \left(evidence\right)}$ In the context of the Monty Hall problem, the hypothesis (h) is that the car is behind the door the contestant originally chose, and the evidence (e) is that the host opens a nonselected door to reveal a goat.

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Applying the theorem:

• P(h) = 1/3​: There is a 1 in 3 chance the car is behind the door originally chosen.
• P(e∣h) = 1: If the contestant chose correctly, the host still has two goat doors to choose from, and will always reveal a goat.
• P(e) = 1: The host always reveals a goat.

Substituting these into the formula:$P (car is behind original choice\mid goat revealed)=\frac{1\times \frac{1}{3}}{1}=\frac{1}{3}$The probability that the car is behind the originally chosen door remains 1/3 even after the host opens a goat door. Therefore, the probability that the car is behind the other unopened door is 2/3, and switching is the better strategy.

Psychological factors at play

The Monty Hall problem sheds light on common cognitive biases, which are systematic errors and limitations in the way individuals reason about the world, and emotional choice biases, wherein feelings and impulses influence logical decision-making. Cognitive biases include the equiprobability bias, which makes contestants erroneously assume that after the host opens one of the three doors the remaining doors carry an equal (50:50) chance of concealing the car. They may also fail to recognize that the host’s choice of which door to open is constrained by the rules of the game: The host knows where the car is and must open a door concealing a goat. Another cognitive limitation commonly at play is the failure of the contestants to recognize the combined probability of the set or group of nonselected doors, mentally assigning individual probabilities to each instead.

Counterfactual thinking, which involves anticipating the regret that may follow if the contestant loses after switching doors, gives rise to “switch aversion,” making contestants more likely to stick with their initial choice. Contestants may also develop a sense of “ownership” over their initial choice, making them unwilling to change it—a tendency experts associate with the endowment effect and status quo bias. An illusion of control may also influence contestants to stick with their initial choice, many overestimating their ability to identify the winning door and also perceiving it as safer, special, or lucky.

From puzzle to public debate

The Monty Hall problem traces its origins to early probability puzzles, which sparked similar interest for their seemingly improbable answers. In 1959 a version of the Monty Hall problem involving three prisoners appeared in American mathematics writer Martin Gardner’s “Mathematical Games” column in the Scientific American. American statistician Frederick Mosteller included a variation in his book Fifty Challenging Problems in Probability with Solutions (1965).

English evolutionary biologist John Maynard Smith presented a version of the puzzle in his book Mathematical Ideas in Biology (1968). Recalling a 1966 theoretical biology conference at Villa Serbelloni in Italy, he remarked:

This should be called the Serbelloni problem since it nearly wrecked a conference on theoretical biology in the summer of 1966; it yields at once to common sense or to Bayes’ theorem.

The game-show version was introduced in 1975 by American statistician Steve Selvin in The American Statistician. His explanation showed that switching improves the odds and led to letters from readers who found the result hard to accept. Public debate surged in 1990 when Parade magazine columnist Marilyn vos Savant—considered to have the highest IQ (228) at the time—answered a reader’s question about the Monty Hall problem. She explained that switching improves the odds of winning, moving the chance from 1/3 to 2/3. Her answer sparked national debate. Many academics insisted that she was mistaken. Several follow-up columns were written following the incident, and classroom experiments were performed to test the validity of vos Savant’s answer. The New York Times even published a front-page article covering the controversy in 1991. Among the skeptics was Hungarian mathematician Paul Erdős, whose doubts about the answer persisted until computer simulations confirmed the result.