About
Monty Hall Problem is a small public experiment. One person plays the puzzle a hundred times. We record every game. Then we show that person their own numbers, next to everyone else's.
What is the Monty Hall problem?
You pick one of three doors. A door you did not pick opens and shows a goat, never the car. Then you choose to stay or switch. Your first pick is right 1 time in 3, so the other door is right 2 times in 3.
The puzzle is named after Monty Hall. He hosted an American game show called Let's Make a Deal, which began in 1963. Steve Selvin wrote the puzzle down in a 1975 letter to a statistics journal. It became famous in 1990, when Marilyn vos Savant answered it in a magazine column and told readers to switch. About ten thousand people wrote in to say she was wrong, and around a thousand of them had doctorates. Paul Erdos, one of the great mathematicians of the last century, is said to have stayed unconvinced until someone showed him a computer simulation.
Why is it worth knowing?
The puzzle is small, but the mistake behind it is not. People judge the odds from how things look right now, and ignore how the situation came about. Two doors are shut, so it looks even. What gets missed is that the host chose which door to open, and he chose it knowing where the car was.
That habit is worth breaking, because it turns up well outside game shows. When someone shows you a result, the useful question is what else they could have shown you. A test result, a number in an argument, a chart in a pitch, a set of survey answers: each one means something different depending on how freely the person could have picked a different one. If they had no choice, it tells you a lot. If they picked it out of many, it tells you much less.
It is also a lesson in how confidently people get probability wrong. Around a thousand readers with doctorates wrote in to tell Marilyn vos Savant she was mistaken. They were sure of themselves, they were qualified, and they were wrong. What settled it was not a better argument. It was playing the game many times and looking at the count, which is what this site is for.
Is it better to switch doors?
Yes. Switching wins 2 times in 3. Switching wins in exactly the games where your first pick was wrong, and your first pick is wrong 2 times in 3.
The step people skip is that the host is not guessing. He knows where the car is, and he never opens that door. So the door he leaves shut carries the odds your first pick missed. Change that one rule and the answer changes. If a door opened at random and happened to show a goat, the two shut doors really would be an even bet. This site plays the version where the host knows, which is the version the puzzle is about.
Why play a hundred games?
One game proves nothing, and ten games can go any way. A hundred games is the point where the two competing answers stop overlapping, so a single run lands clearly nearer one of them.
Put numbers on it. If switching really wins 2 times in 3, your first pick should be wrong about 67 times in 100. If the two shut doors were really an even bet, it would be about 50. Over ten games those two predictions are only about 7 and 5, close enough that ordinary luck covers the gap, and almost any result fits both stories. Over a hundred games they are 67 and 50, which is more than three standard deviations apart. A typical run lands between 58 and 76. That is why the count has to be this long: shorter runs cannot tell the two answers apart, and you would be back to arguing.
What we measure
We count one thing: the first pick, and how often it was not the car. That is the same as asking how often switching would have won. It does not depend on how you play. Someone who always stays and someone who always switches produce the same number here.
- For each player, how many of their hundred first picks were wrong. Across everyone, these should sit under a Binomial(100, 2/3) curve centred on 67.
- Across every game ever played, the share of first picks that were wrong. This should settle on two thirds and stay there.
- Whether a run was finished or left early. Finished and unfinished runs are reported separately. We never merge them.
How we set the doors
The server places the car before you pick. Your browser is not told where the car is until after you stay or switch. The door that opens is picked from the doors that are neither your pick nor the car. If your pick happens to be the car, either of the other two is equally likely to open. The randomness comes from Go's ChaCha8 generator, seeded by the operating system when the server starts.