Monty Hall problem
What is the Monty Hall problem?
Why does switching doors increase the chances of winning in the Monty Hall problem?
How does the Monty Hall problem illustrate conditional probability?
What are some cognitive biases related to the Monty Hall problem?
How did the Monty Hall problem gain public attention?
Imagine you are a contestant on a game show. You are presented with three closed doors. Behind one is a car—the prize you want. Behind the other two are goats. You choose one door, but it remains closed. Now the host—who knows what is behind each door—opens one of the other two doors, revealing a goat. You are then offered a choice:
- Stay with your original choice or
- Switch to the other unopened door.
Which option would you choose to maximize your chances of winning the car?
This is the Monty Hall problem, a probability puzzle named after Monty Hall, the host of the American television game show Let’s Make a Deal (1963–2003). Based on a scenario from the show, it presents a seemingly simple choice—but with a solution that defies common intuition. First posed in print by statistician Steve Selvin in 1975, the Monty Hall problem gained attention in the 1990s and is now widely cited in mathematics, decision theory, and cognitive psychology.
The answer
It may seem as though the odds are now 50–50 between the two remaining doors, but they are not. Switching your choice to the other unopened door increases your chances of winning to 2/3, while staying with your original choice leaves you with just a 1/3 chance. This result has been verified both mathematically and through repeated simulations. The reason why lies in understanding how Monty’s knowledge and actions influence the probability landscape.
Why switching works
Initial probabilities
When a contestant chooses one door out of three:
- The probability that the car is behind the contestant’s chosen door is 1/3.
- The probability that the car is behind one of the other two doors is 2/3.
Host’s action
Monty opens one of the two doors not initially selected, revealing a goat. This step is not random—he knows where the car is and never chooses to open that door. This action gives the contestant additional information about the choices on the table. One losing option has been removed from the two-door group of doors that the contestant did not choose, which originally had a 2/3 chance of containing the car. Since that group now consists of only one unopened door, the entire 2/3 chance transfers to it.
Stick or switch
- Staying with the original pick gives the contestant a 1/3 chance of winning.
- Switching to the other unopened door gives the contestant a 2/3 chance.
Thus, over many repetitions of the game, switching wins twice as often as staying.
A table of outcomes
The logic becomes clearer with more doors. Suppose there are 100 doors. You pick one. Monty opens 98 others, all showing goats, leaving one unopened door besides your original choice.
- Your original pick had a 1/100 chance of being right.
- The remaining unopened door has a 99/100 chance.
Would you switch? Most people would—revealing how intuition improves when the asymmetry becomes more obvious.
Assume the contestant always picks door 1:
| where the car is | host reveals | stay | switch |
|---|---|---|---|
| door 1 | door 2 or 3 | win | lose |
| door 2 | door 3 | lose | win |
| door 3 | door 2 | lose | win |
Only in one of the three equally likely cases does staying with the original choice result in a win. Switching leads to a win in two out of three cases.
Mathematics behind the puzzle
The Monty Hall problem is a canonical illustration of conditional probability—the principle that the likelihood of an event can change when additional relevant information is revealed. It serves as an accessible entry point to Bayesian reasoning, a concept in probability theory.
Bayes’s theorem allows us to update the probability that the car is not behind the contestant’s initial choice of doors, given that Monty opens a door to reveal a goat. The theorem states: Let’s define the terms in this context:
- Hypothesis (h): The car is behind your original choice.
- Evidence (e): Monty opens another door to reveal a goat.
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, Monty still has two goat doors to choose from, and will always reveal a goat.
- P(e) =1: Monty always reveals a goat.
Substituting these into the formula:The probability that the car is behind the originally chosen door remains 1/3 even after Monty 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.
From puzzle to public debate
A reference to the Monty Hall problem appears in the comedy series Brooklyn Nine-Nine (season four, episode 8), where Captain Holt and his husband, Kevin, debate the puzzle’s logic, and Detective Amy Santiago confirms that switching doors improves the odds of winning.
Even after seeing the math, many people hesitate to switch doors. The Monty Hall problem isn’t just a probability puzzle but also an illustration of common cognitive biases (systematic errors in the way individuals reason about the world).
- Counterfactual thinking: People often imagine the regret of switching and then losing (“If only I had stayed”), which discourages switching and strengthens the stay bias.
- Illusion of control: Sticking with your first choice feels safer—even when it isn’t.
- Equiprobability bias: Once a door is opened, many assume the remaining options are equally likely. They’re not. This mistaken belief comes from how we learn randomness—fair coins, dice, and cards—where outcomes are equal. But Monty’s choice isn’t random; he knows where the car is, which shifts the odds.
Psychologists such as Donald Granberg and Thomas Gilovich have studied these effects in depth. The takeaway: Intuition misfires, probability wins.
In 1959 a version of the Monty Hall problem involving three prisoners appeared in mathematics writer Martin Gardner’s “Mathematical Games” column in the Scientific American. Statistician Frederick Mosteller included a variation in his book Fifty Challenging Problems in Probability with Solutions (1965), and stated that the problem attracted more letters from readers than any other.
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 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 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 and drew thousands of letters, including many from academics who insisted that she was mistaken. The controversy prompted follow-up columns, classroom experiments, and even a front-page article in The New York Times in 1991. Among the skeptics was mathematician Paul Erdős, who reportedly maintained his doubts until computer simulations confirmed the result.