The Monty Hall Problem Explained (and Why You Should Switch)

The Monty Hall problem is the most famous counterintuitive puzzle in all of probability, the one that feels wrong even after you've proven it's right. Here's the setup: you're on a game show facing three doors. Behind one is a car; behind the other two are goats. You pick a door. The host, who knows what's behind each one, opens a different door to reveal a goat, then asks: do you want to switch to the other unopened door, or stay? Almost everyone says it doesn't matter, it's 50/50. They're wrong. Switching wins two-thirds of the time. This guide explains why, in three different ways, until it finally clicks.

The puzzle, precisely

The exact rules matter, so let's state them:

1. Three doors: one hides a car, two hide goats.
2. You pick a door (say Door 1), but it stays closed.
3. The host, who knows where the car is, opens one of the other two doors, always revealing a goat.
4. The host then offers you the choice to switch to the remaining closed door or stay with your original pick.

The question: to maximize your chance of winning the car, should you stay or switch?

The answer: always switch

Switching wins the car 2/3 of the time. Staying wins only 1/3 of the time. This isn't a trick of wording, it's a real, provable fact, and it's one of the best examples of why your gut can't be trusted on probability. Let's see why.

Why it works: the intuition

The key is what happens at the start. When you first pick, you have a 1 in 3 chance of choosing the car. That means there's a 2 in 3 chance the car is behind one of the other two doors.

Now the host opens a goat door, but here's the crucial part: the host knows where the car is and deliberately avoids it. He's not opening a random door, he's giving you information. That 2/3 probability that the car was "over there" doesn't vanish; it gets concentrated entirely onto the single door he didn't open. So the remaining door carries the full 2/3 chance, while your original pick still carries only 1/3. Switching grabs the bigger share.

Why it works: proof by cases

Don't trust the hand-waving? Just count. You always pick Door 1. The car is equally likely behind each door:

• Car behind Door 1 (chance 1/3): the host opens Door 2 or 3 (a goat). If you switch, you get a goat, you lose. Stay wins.
• Car behind Door 2 (chance 1/3): the host must open Door 3 (the only goat he can show). If you switch to Door 2, you win the car.
• Car behind Door 3 (chance 1/3): the host must open Door 2. If you switch to Door 3, you win the car.

Tally it up: switching wins in two of the three equally likely cases. That's a 2/3 win rate, exactly as promised. Staying wins in only one case, 1/3.

Why it works: the 100-door trick

If it still feels wrong, scale it up. Imagine 100 doors, one car, 99 goats. You pick one door, your chance of being right is a measly 1/100. Now the host, who knows where the car is, opens 98 other doors, all goats, leaving just your door and one other. Do you switch?

Of course you do. There was a 99/100 chance the car was not behind your first pick, and the host has helpfully cleared away every wrong door except one. That remaining door has a 99/100 chance of hiding the car. The three-door version is the same effect, just smaller and easier to dismiss.

The history: a famous controversy

The problem is named after Monty Hall, host of the game show Let's Make a Deal. It became famous in 1990 when columnist Marilyn vos Savant answered it in Parade magazine and said to switch. The backlash was enormous, she received thousands of letters insisting she was wrong, including some from people with PhDs in mathematics. But she was right, and computer simulations and formal proofs settled it decisively. It remains a humbling reminder that probability fools even experts.

The lesson for every brain teaser

The Monty Hall problem is the perfect illustration of the most important brain teaser skill: distrust your intuition on probability. The "obvious" 50/50 answer comes from treating the host's action as random, when in fact it's loaded with information. Whenever a puzzle involves someone who knows the answer making a choice, stop and ask what their choice reveals. That single habit, covered more in how to solve brain teasers, cracks a huge class of counterintuitive puzzles.

Want more puzzles that punish your first instinct? The Monty Hall problem lives at our expert brain teasers level, alongside other counterintuitive stumpers, and its cousin in surprising probability is the birthday paradox.

Frequently asked questions

What is the Monty Hall problem?

It's a probability puzzle based on a game show. You pick one of three doors (one hides a car, two hide goats), the host opens a different door to reveal a goat, and you choose whether to switch to the last door or stay. The surprising answer is that switching wins two-thirds of the time.

Why should you switch in the Monty Hall problem?

Because your first pick has only a 1/3 chance of being the car, leaving a 2/3 chance it's elsewhere. The host knowingly opens a goat door, concentrating that entire 2/3 probability onto the one remaining door. Switching therefore wins 2/3 of the time, while staying wins just 1/3.

Is the Monty Hall problem really 2/3?

Yes. It has been proven mathematically and confirmed by countless computer simulations. The result depends on the host always knowingly revealing a goat and always offering the switch. Under those standard rules, switching wins exactly two-thirds of the time.

Why does the Monty Hall answer feel wrong?

Because our intuition treats the two remaining doors as a fresh 50/50 choice, ignoring that the host's reveal wasn't random, it was informed. The host actively avoided the car, which transfers information to the unopened door. Once you account for the host's knowledge, the 2/3 result follows.