Monty Hall Problem
Simulate the Monty Hall puzzle online. Compare stay vs switch with Monte Carlo simulation. See why switching gives a 2/3 chance of winning a car.
About This Calculator
The Monty Hall Problem is a famous probability puzzle named after the host of the television game show "Let's Make a Deal." You are presented with three closed doors — one hides a car, and two hide goats. After you pick a door, the host, who knows what is behind each door, opens a different door that reveals a goat. You are then given the option to stick with your original choice or switch to the other unopened door.
Counterintuitively, switching doors gives you a 2/3 probability of winning the car, while staying gives only a 1/3 probability. This result defies most people's intuition, which assumes that two remaining doors each have a 50% chance. The key insight is that the host's action of revealing a goat provides additional information that changes the probabilities based on conditional probability and Bayes' theorem.
This calculator uses Monte Carlo simulation to run thousands of randomized trials of the Monty Hall game. You can customize the number of doors and the number of simulations to observe how the law of large numbers causes the simulated results to converge toward the theoretical probabilities. With more doors, the advantage of switching becomes even more pronounced — with 10 doors, switching wins 90% of the time versus just 10% for staying.
The Monty Hall Problem was popularized by Marilyn vos Savant in her Parade magazine column in 1990. Her correct solution generated thousands of letters from readers — including many PhDs in mathematics — who insisted she was wrong. This controversy made the Monty Hall Problem one of the most famous examples of how human intuition can fail when evaluating probabilities, and it remains a staple of probability theory and statistics education worldwide.
Regional Notes
Global: The Monty Hall Problem is a universal probability puzzle that applies regardless of region. The theoretical probabilities (1/n for staying, (n-1)/n for switching with n doors) are mathematically proven and do not vary by country. This calculator uses the same simulation logic worldwide.
Frequently Asked Questions
What is the Monty Hall Problem?
The Monty Hall Problem is a probability puzzle based on a game show where a contestant chooses one of three doors, one hiding a car and two hiding goats. After the initial pick, the host (Monty Hall) opens a different door revealing a goat, then offers the contestant a chance to switch to the remaining unopened door. The puzzle demonstrates that switching doors gives a 2/3 chance of winning the car, while staying gives only a 1/3 chance.
Why is the Monty Hall Problem not 50/50?
After the host opens a goat door, many assume the two remaining doors each have a 50% chance. However, the initial choice had a 1/3 probability of being correct, and this probability does not change when the host reveals a goat. The other unopened door inherits the combined 2/3 probability from the two doors the contestant did not initially pick, making switching twice as likely to win.
Should you switch doors in the Monty Hall Problem?
Yes, you should always switch doors. The probability of winning the car by switching is 2/3 versus 1/3 by staying. This is because when you initially pick a door with a goat (which happens 2/3 of the time), the host is forced to reveal the other goat, leaving the car behind the only remaining door.
How does the Monty Hall Problem work with more than 3 doors?
With n doors where the host opens all goat doors except one, the probability of winning by staying is 1/n, while switching gives (n-1)/n chance of winning. For example, with 10 doors, staying wins only 10% of the time while switching wins 90% of the time, making the advantage of switching even more dramatic.
Who solved the Monty Hall Problem?
Marilyn vos Savant, listed in Guinness World Records for the highest IQ, first published the correct solution in her Parade magazine column in 1990. She explained that switching doors doubles the probability of winning. Her answer sparked widespread debate, with thousands of readers including mathematics professors initially insisting she was wrong, before eventually being proven correct.
How does the Monte Carlo simulation work in this calculator?
This calculator runs thousands of randomized trials of the Monty Hall game. In each trial, the car is placed behind a random door and the contestant picks a random door. If they guessed correctly, staying wins. If they guessed incorrectly, switching wins. After all trials, the win rates for both strategies are computed and compared against the theoretical probabilities.
What happens if you increase the number of doors in the Monty Hall Problem?
Increasing the number of doors makes switching even more advantageous. With 3 doors, switching gives 66.7% win rate. With 10 doors, switching gives 90% win rate. With 100 doors, switching gives 99% win rate. The stay strategy always has exactly 1/n probability regardless of n, while switching has (n-1)/n probability.
Can the Monty Hall Problem be applied to real life?
Yes, the Monty Hall Problem illustrates the important concept of conditional probability and how new information should update our beliefs. It has applications in decision theory, statistics, Bayesian inference, and understanding how probabilities shift when information is revealed. It serves as a cautionary example against relying on intuition for probabilistic reasoning.