The Monty Hall Problem, Explained So It Finally Clicks

The Monty Hall problem is famous for two reasons. The answer is simple, and almost nobody believes it the first time.

You’re on a game show. There are three doors. Behind one is a car, and behind the other two are goats. Here’s how it goes:

1. You pick a door, say door 1.
2. The host, who knows where the car is, opens one of the other doors, say door 3, and shows you a goat.
3. He offers you a choice: stick with door 1, or switch to door 2.

Should you switch?

The answer

Yes. Switching wins the car 2 times in 3. Sticking wins only 1 time in 3.

If you feel a strong pull towards “it’s 50/50”, you’re in good company. There are two doors left and one car, so it seems like a coin flip. When the columnist Marilyn vos Savant published the correct answer in Parade magazine in 1990, she received thousands of letters telling her she was wrong, and many were from people with PhDs. The mathematician Paul Erdős is said to have refused to accept it until a computer simulation showed him.

Pigeons, as it happens, do better than most people. In a 2010 experiment, Walter Herbranson and Noam Schroeder gave pigeons a version of the game with food behind the doors. After a month of practice the birds had worked out that switching pays. Human volunteers on the same task mostly kept sticking.

So here are four ways to see it. Stop reading when one of them clicks.

Way 1: your first pick doesn’t improve

When you first pick a door, you have a 1 in 3 chance of choosing the car. Nothing the host does can change that, because he’ll always open an empty door, whichever one you picked. So your door is still a 1 in 3 shot.

The other 2 in 3 has to go somewhere. It was spread across the two other doors, and now the host has told you which of those is empty. All of it now sits on the one door he left closed.

Way 2: list every case

Say the car is behind door 1, 2 or 3 with equal chance, and you always pick door 1. There are three equally likely worlds:

Sticking wins in one world out of three. Switching wins in two. The two bottom rows are the key: when your first pick was wrong, which happens 2 times in 3, the host has no choice. He must open the one remaining goat door, and the door he leaves shut is the car.

Way 3: a hundred doors

Make it 100 doors. You pick door 1. The host, who knows where the car is, opens 98 of the other doors, all of them goats, and leaves one closed, door 59. Do you stick or switch?

Almost everyone says switch, and it’s clear why. Your door had a 1% chance when you picked it. The host cleared 98 doors and carefully skipped one. That door is far more likely than yours to hide the car. Three doors is the same puzzle, just with the numbers small enough to disguise it.

Way 4: just play it

If none of those works, play it. The toy at the top lets you run the game, or run it a hundred times each way. After a few dozen rounds you’ll see sticking win about a third of the time and switching about two thirds. It’s a hard result to argue with.

Why it feels like 50/50

The pull towards 50/50 comes from a natural-seeming rule: when you’re left with two options and no reason to prefer either, treat them as equal. It works for a coin. It doesn’t here because the two doors aren’t symmetrical. One is the door you picked, before any information arrived. The other is the door the host chose not to open, which tells you something.

There’s also a psychological pull. If you stick and lose, you picked the car and gave it up by your own choice. If you switch and lose, you gave up the car by switching. Research on the problem found that people who switch and lose regret it more than people who stick and lose, even though switching is the better bet. Many people stay put to avoid the regret.

The rules matter

The puzzle depends on three assumptions that are easy to miss, because the original wording leaves them implied:

• The host knows where the car is.
• The host always opens a door with a goat.
• The host always offers you the switch.

Change any of these and the answer changes. The famous variation is the Monty Fall problem: the host stumbles, bumps a door and it swings open to reveal a goat. He didn’t know it was a goat. If the car is still hidden, the odds really are 50/50, because his action carried no information. The difference is whether the host’s choice was constrained by what’s behind the doors.

It’s also worth saying that the real Monty Hall didn’t play it exactly this way. He didn’t always offer a switch, and sometimes he offered it precisely when you’d picked the car, to tempt you away. The puzzle is about an idealised version, and its lesson stands: information changes odds, but only when it’s tied to something real.

How to explain it to someone who won’t budge

Almost everyone ends up in this conversation at some point, so it’s worth having a script.

Don’t start with the maths. Start with the hundred doors and ask what they’d do. When they say switch, ask what changed between that version and the three-door one. The honest answer is only the numbers. Then ask them to imagine you’re the one picking, and they’re the host who knows where the car is: you pick door 1, they open the rest, and they have to leave exactly one closed. If you picked wrong, they are forced to leave the car shut. That happens two times in three.

If they still won’t move, play it with three playing cards, one of them an ace. Do it twenty times each way. Nothing argues like a tally.

A short history of a famous argument

The puzzle is older than Marilyn vos Savant. In 1975 the statistician Steve Selvin posed it in a letter to the journal The American Statistician, naming it after the host of the TV game show Let’s Make a Deal. It lay quiet for fifteen years. Then a reader wrote to Parade magazine in 1990, vos Savant gave the right answer in her column, and the post bags filled with fury.

What makes the story good isn’t that the readers were silly. Many of the people who wrote in understood probability well. The puzzle works because a perfectly reasonable rule (equal options mean equal chances) fails for a reason that is easy to overlook (the host’s choice depends on the car). That’s a more common situation than a game show.

What it teaches

The Monty Hall problem is a small, clean example of conditional probability: how the chance of something changes once you learn something new. The surprising part is that new information doesn’t always help in the way you’d expect. The host’s reveal didn’t improve your door’s odds. It sent all the remaining chance to the other door.

You can see the same shape in medicine, in detective stories and in everyday decisions. A test result, a clue or a leaked rumour is only as informative as the process that produced it. The question to ask isn’t just “what did I learn?” but “how did I come to learn it, and could it have turned out differently?”

If you want to try your own reasoning on more puzzles like this, the free probability test has Monty Hall alongside shared birthdays, streaks and medical tests.