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.
Understanding Statistics: The Monty Hall Problem
The puzzle is really about updating information. Before any door is opened, your pick is only one possible location for the prize. The important event is not simply that a goat becomes visible.
It is that the host deliberately chooses a goat door that you did not select. This action carries information because the host follows rules based on knowledge of the prize location.
In probability, this is called conditional probability. It means that the chance of an outcome can change after learning new information, especially when the source of that information has a rule for what it reveals.
A useful way to reason through the game is to list the possible starting situations. In one situation, the first choice finds the car. In the other two situations, the first choice finds a goat.
If the first choice finds the car, changing doors loses. If the first choice finds a goat, the host has no freedom about the remaining unopened door. That door must hide the car, so changing wins.
There are two losing first choices and only one winning first choice. Switching turns both of those losing starts into wins. Staying succeeds only in the single case where the first choice was already right.
The host's rules are essential. Suppose the host sometimes opens a door at random and might reveal the car. Then seeing a goat would need a different calculation.
Suppose the host knows the prize location but sometimes offers a switch only when you chose correctly. Then the offer itself gives extra information. A probability answer is never separate from the process that created the evidence.
When studying problems like this, write down exactly what each person knows, what actions are allowed, and whether an action is chosen randomly or by a fixed rule. Small changes in these conditions can produce very different results.
Real data problems often work the same way. A medical test result depends on how the test was selected, who was tested, and how accurate the test is. A news poll depends on who was asked and who chose to respond.
In each case, observed information may be filtered by a process that is not random. Students often feel that two closed doors must have equal chances because there are two doors left. That shortcut ignores the host's knowledge and choice.
To build confidence, run a simple simulation with cards or numbered slips. Record the first pick, the revealed goat, and the final result over many rounds. The totals make the hidden pattern easier to trust than intuition alone.
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.