Monty Hall Problem — Definition, Formula & Examples
The Monty Hall Problem is a famous probability puzzle where a contestant picks one of three doors, the host reveals a losing door, and the contestant must decide whether to switch. Counterintuitively, switching doors doubles your probability of winning from 1/3 to 2/3.
A conditional probability problem in which a player selects one of three doors (one hiding a prize), the host (who knows what is behind each door) opens a different door that does not contain the prize, and the player is offered the choice to stay or switch. By Bayes' theorem, the posterior probability of the prize being behind the remaining unchosen door is 2/3, while the probability of it being behind the originally chosen door remains 1/3.
How It Works
Suppose you pick Door 1. There is a 1/3 chance you picked correctly and a 2/3 chance the prize is behind one of the other two doors. When the host opens a losing door from the remaining two, that 2/3 probability doesn't disappear — it collapses entirely onto the one unopened door. The host's action gives you new information because the host always knows where the prize is and always reveals a losing door. This is why switching wins 2/3 of the time, not 1/2.
Worked Example
Problem: You are on a game show with 3 doors. Behind one door is a car; behind the other two are goats. You pick Door 1. The host, who knows what is behind each door, opens Door 3 to reveal a goat. Should you switch to Door 2 or stay with Door 1?
Step 1: Find the initial probability that your chosen door (Door 1) has the car.
Step 2: Find the initial probability that the car is behind one of the other doors (Door 2 or Door 3 combined).
Step 3: The host opens Door 3 (a goat). Since the host always reveals a goat, the entire 2/3 probability shifts to Door 2.
Answer: You should switch to Door 2. Switching gives you a 2/3 chance of winning the car, while staying gives you only a 1/3 chance.
Visualization
Why It Matters
The Monty Hall Problem is one of the clearest demonstrations that human intuition about probability can be deeply wrong. It appears in AP Statistics and introductory college probability courses as a gateway to conditional probability and Bayesian reasoning. Understanding it strengthens your ability to analyze any situation where new evidence should update your beliefs.
Common Mistakes
Mistake: Assuming the odds reset to 1/2 after the host opens a door, since only two doors remain.
Correction: The host's action is not random — the host always opens a losing door. This means your original 1/3 vs. 2/3 split is preserved, not replaced by a fresh 50-50 split. The key is that the host's choice depends on where the prize is.