The Monty Hall Problem

You picked a door. The host just opened one. Should you switch?

Three doors. Behind one is a car. Behind the other two are goats. You pick a door — say, Door 1. Before opening it, the host (who knows where the car is) opens a different door to reveal a goat. Now he offers you a choice: stay with Door 1, or switch to the remaining closed door.

Most people say it doesn't matter. Two doors left, one car — it must be 50/50.

They are wrong. Switching wins twice as often as staying.

This result has enraged mathematicians, confused PhD students, and generated thousands of letters of protest. It's also completely correct — and understanding why is one of the cleanest lessons in probability.

The intuition: the host's action is not neutral

Here's the key move. You are not watching a fresh coin flip after the reveal. You are watching the host do something constrained by information you don't have.

Think about what happens from the car's perspective across many games.

Your initial pick is right 1 in 3 times. In those cases, the host can open either of the two remaining goat doors — doesn't matter which. Staying wins.

Your initial pick is wrong 2 in 3 times. In those cases, the host has no choice: he must open the one goat door that isn't yours, because the other remaining door holds the car. Switching wins.

The host's reveal doesn't split the probability evenly between the two remaining doors. It funnels the entire 2/3 probability that your first pick was wrong onto the single door you could switch to.

Try it yourself

Play a few rounds by hand, then hit "Run 1,000 trials" to watch the rates converge. The bars settle near the theoretical values: Stay ≈ 33.3%, Switch ≈ 66.7%.

Monty Hall simulator

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Door 1

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Door 2

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Door 3

Pick a door to start.

Run 1,000 trials

Notice that the first few games feel random — you can easily win twice in a row by staying. The law of large numbers takes a hundred or more games to reveal the underlying structure. This is exactly why our intuition fails: single outcomes swamp signal.

The formal argument

Setup: one car, two goats, three doors. Car placed uniformly at random.

$P(\text{car behind door } i) = \tfrac{1}{3} \quad \text{for } i = 1, 2, 3$

Stay strategy: you win if and only if your initial pick was correct.

$P(\text{win} \mid \text{stay}) = P(\text{initial pick correct}) = \frac{1}{3}$

Switch strategy: you win if and only if your initial pick was wrong — because in that case the host is forced to reveal the only goat he can, leaving the car behind the remaining door.

$P(\text{win} \mid \text{switch}) = P(\text{initial pick wrong}) = \frac{2}{3}$

Switching doubles the win probability from 1/3 to 2/3.

The Bayes view: the cleanest way to see it is to condition on the specific door the host opens. Say you picked Door 1 and the host opens Door 3 — so Door 2 is the door you'd switch to. Which door the host opens depends on where the car is:

• If the car is behind Door 1 (your pick, prior $\tfrac{1}{3}$): the host may open Door 2 or Door 3 and picks at random, so $P(\text{opens 3}) = \tfrac{1}{2}$.
• If the car is behind Door 2 (prior $\tfrac{1}{3}$): the host is forced to open Door 3, so $P(\text{opens 3}) = 1$.
• If the car is behind Door 3 (prior $\tfrac{1}{3}$): the host cannot open it, so $P(\text{opens 3}) = 0$.

The total chance the host opens Door 3 is therefore $\tfrac{1}{3}\cdot\tfrac{1}{2} + \tfrac{1}{3}\cdot 1 + \tfrac{1}{3}\cdot 0 = \tfrac{1}{2}$. Now apply Bayes' theorem for the switch door:

$P(\text{car behind 2} \mid \text{opens 3}) = \frac{P(\text{opens 3} \mid \text{car behind 2})\,P(\text{car behind 2})}{P(\text{opens 3})} = \frac{1 \cdot \tfrac{1}{3}}{\tfrac{1}{2}} = \frac{2}{3}$

The posterior climbs to 2/3 because the host opening Door 3 is twice as likely when the car is behind Door 2 (he's forced to) as when it's behind your Door 1 (a coin flip). That asymmetry — the information carried by the host's constrained choice — is exactly what the 50/50 intuition throws away.

Why this matters beyond game shows

The Monty Hall problem is not a curiosity. It is the simplest possible illustration of a principle that breaks real analyses:

The data-generating process is part of the probability. How information reaches you changes what that information means. The host's reveal looks like "one fewer door" but it is not a random reveal — it is a constrained one. Ignoring that constraint produces the wrong answer.

This shows up everywhere:

• Survival bias in datasets: you only observe entities that survived a selection process. The selection process isn't neutral. See base rate neglect for a related failure mode.
• A/B test stopping rules: peeking at results and stopping when you see significance changes the probability that significance is real — the stopping rule is part of the data-generating process.
• Bayesian updating in general: Bayes' theorem exists precisely to keep the mechanism by which evidence arrived inside the calculation. The Monty Hall problem is Bayes' theorem with the training wheels off.

Whenever you see a probability that seems obvious, ask: what constraints shaped how this information reached me? The answer is usually not neutral.