The Monty Hall problem, explained
You are on a game show, facing three closed doors. Behind one is a car. Behind the other two are goats. You pick a door. The host, who knows where the car is, opens one of the other two doors to reveal a goat. Then he makes you an offer: stick with your first choice, or switch to the one remaining closed door.
Almost everyone says it makes no difference: two doors are left, so it must be 50/50. It is not. Switching wins two times out of three. Play a few rounds below, then let the simulator play ten thousand games each way.
Try it yourself
One door hides a car, the other two hide goats. Pick a door.
- Switch: won 0 of 0–
- Stick: won 0 of 0–
Why switching wins two times in three
Your first pick has a 1 in 3 chance of being the car. Nothing the host does afterwards can change that, because he can always open a goat door whatever you picked. So one time in three you start on the car, and switching loses. Two times in three you start on a goat, the host is forced to open the only other goat door, and the door he leaves closed is the car. Switching wins.
Sticking wins only when your first guess was right: 1 in 3. Switching wins whenever your first guess was wrong: 2 in 3.
| Car is behind | You pick | Host opens | If you stick | If you switch |
|---|---|---|---|---|
| Door 1 | Door 1 | Door 2 or 3 | Win | Lose |
| Door 2 | Door 1 | Door 3 | Lose | Win |
| Door 3 | Door 1 | Door 2 | Lose | Win |
The table shows every case when you start on door 1. The three rows are equally likely, and switching wins in two of them. Starting on door 2 or 3 gives the same pattern.
The hundred-door version
If the argument still feels wrong, make the game bigger. Imagine 100 doors, one car and 99 goats. You pick door 17. The host, who knows where the car is, opens 98 other doors, every one a goat, and leaves just door 62 closed.
Would you stick with door 17? Your first pick had a 1 in 100 chance. The host has just walked past 98 doors and carefully avoided one. Unless you were lucky enough to pick the car straight away, which happens one time in a hundred, door 62 is the car. Switching wins 99 times in 100. The three-door game is the same thing on a smaller scale, where the host's help is less obvious.
Why it feels like 50/50
The mistake is to treat the two remaining doors as if they had just been put in front of you. A newcomer who walked in at that moment and chose between them with a coin flip really would win half the time. But you are not a newcomer. You know something about how those two doors came to be the last ones standing.
The host's choice was not random. He was not allowed to open your door, and he was not allowed to reveal the car. When your first pick was a goat, that left him exactly one door he could open. His action carries information about the door he avoided, and none at all about the door you chose, which he could never have opened anyway. That lopsided information is the entire puzzle.
The rules that make it work
The 2 in 3 answer depends on the host behaving in a particular way. Change his behaviour and the answer changes too.
- The host knows where the car is.
- He always opens a door, and always one hiding a goat.
- He never opens the door you picked.
- He always offers the switch, whether or not your first pick was right.
Suppose instead the host has no idea where the car is, opens one of the other doors at random and just happens to reveal a goat. Now the odds really are 50/50. His reveal no longer tells you anything special, because he might just as easily have exposed the car. And a host who only offers the switch when you have picked the car would make switching a certain loss. The simulator on this page follows the standard rules listed above.
The puzzle that fooled the experts
The problem takes its name from Monty Hall, the long-serving host of the American game show Let's Make a Deal. The statistician Steve Selvin posed it in a letter to the journal The American Statistician in 1975, but it became famous in 1990, when Marilyn vos Savant answered a reader's version of it in her column in Parade magazine. She said you should switch.
Her answer drew around ten thousand letters, close to a thousand of them from people with doctorates, and the great majority told her she was wrong, some in very condescending terms. She was right. Even Paul Erdős, one of the most prolific mathematicians of the twentieth century, is reported to have refused to accept the answer until he was shown a computer simulation of the game.
That is worth remembering when your own instinct rebels. This is a problem where clever, trained people confidently get the wrong answer, and where running the experiment is more persuasive than any argument.
What the puzzle teaches
The lesson is that probability depends on what you know and on how you came to know it. The same two closed doors carry different odds for you and for a newcomer, because you watched the host make a constrained choice. Statisticians call this conditional probability, and it turns up whenever information reaches you through a filter: a medical test that is only given to people with symptoms, a news story selected because it is unusual, a product review written mostly by people who were delighted or furious.
It is also a reminder that a fair random device has no such memory or motive. A wheel does not know which slices you are hoping for and does not adjust. If you want to see what plain, uninformed randomness looks like, put three entries on the random name picker, open the Odds tab and run the fairness test: each one wins a third of the time, with no host steering the result. For the opposite mistake, expecting a random device to remember, see our guide to the gambler's fallacy.
Frequently asked questions
Should you switch doors in the Monty Hall problem?
Yes. Under the standard rules, switching wins the car two times in three and sticking wins one time in three.
Why is the Monty Hall problem not 50/50?
Because the host knows where the car is and must open a goat door other than yours. Your first pick keeps its 1 in 3 chance, so the other closed door carries the remaining 2 in 3.
What if the host opens a door at random?
If the host does not know where the car is and reveals a goat only by luck, the two remaining doors are equally likely and switching makes no difference.
Who was Monty Hall?
Monty Hall was the host of the American television game show Let's Make a Deal, which began in 1963. The puzzle is loosely based on the choices contestants faced on the show.
Has the answer been tested?
Yes, many times, by computer simulation and by playing the game with cards or cups. Switching wins about 66.7% of the time. The simulator on this page lets you run 10,000 games each way.