The Monty Hall problem is a famous probability puzzle based on a game show with three doors, one car, and two goats. You choose one door, then the host opens a different door that always contains a goat. You are then offered a choice to stay with your original door or switch to the other unopened door.
The surprising result is that switching wins the car with probability 2/3, while staying wins with probability 1/3.
The key idea is that the host's action is not random in the usual sense because the host knows where the car is and must reveal a goat. Your first choice has a 1/3 chance of being correct and a 2/3 chance of being wrong. If your first choice was wrong, switching moves you to the car because the host has removed the only other goat.
Simulations confirm this pattern: over many trials, switching wins about twice as often as staying.
Key Facts
- Initial probability your chosen door has the car: P(car behind chosen door) = 1/3
- Initial probability the car is behind one of the two unchosen doors: P(car behind unchosen doors) = 2/3
- After the host opens a goat door, staying keeps the original probability: P(win by staying) = 1/3
- After the host opens a goat door, switching gets the combined unchosen probability: P(win by switching) = 2/3
- Expected wins in n games by staying: wins = n/3
- Expected wins in n games by switching: wins = 2n/3
Vocabulary
- Conditional probability
- The probability of an event given that some other event is known to have happened.
- Sample space
- The set of all possible outcomes in a probability situation.
- Host information
- The knowledge used by the host to choose a door that hides a goat and is not your selected door.
- Simulation
- A repeated experiment or computer model used to estimate probabilities by counting outcomes.
- Strategy
- A fixed rule for making choices, such as always staying or always switching.
Common Mistakes to Avoid
- Thinking the final two doors are equally likely, because the host's choice gives information and does not reset the original probabilities.
- Treating the host as if they opened a random door, because in the standard Monty Hall problem the host always avoids the car and always opens a goat door.
- Forgetting that the original chosen door keeps probability 1/3, because no new random event has moved the car or changed your first pick.
- Running too few simulation trials, because small samples can fluctuate and hide the long-run pattern that switching wins about 2/3 of the time.
Practice Questions
- 1 In 300 Monty Hall games using the always-switch strategy, about how many wins should you expect?
- 2 In 90 Monty Hall games using the always-stay strategy, about how many wins and losses should you expect?
- 3 Explain why the host opening a goat door does not make the two remaining unopened doors equally likely to hide the car.