Three doors / 200 rounds / two strategies

Monty Hall Simulation

The argument never convinces anyone. The count does. Press run: the first round plays at human speed, the next few speed up, and then it stops being watchable and just runs. Every round scores both strategies, because every round settles both at once — if staying would have won, switching would have lost, and the other way around.

MONTY

YOU

Three doors. One car, two goats. Press run and watch it happen 200 times.

The dot's policy

This only changes which door the green dot walks to. The scoreboard tallies both strategies either way.

Scoreboard

Rounds played: 0

If you stay

—

no rounds yet

If you switch

—

no rounds yet

Nothing has been run yet. The theoretical answer is 1/3 for staying and 2/3 for switching — the point of this page is that you don't have to take that on faith.

Erdős and the 100,000 runs

Paul Erdős — one of the most prolific mathematicians who ever lived — did not believe switching was better. Andrew Vázsonyi ran a computer simulation 100,000 times and showed him the output. Stay won about a third. Switch won about two thirds. Erdős accepted the numbers and still said he did not understand why. Vázsonyi showed him a decision tree. It didn't land. He wanted a reason, not a diagram.

So here's the same demonstration Vázsonyi had, minus the mainframe. If 200 rounds still feels like it could be a fluke, run the same 100,000 he did.

Car is equally likely behind each door1/3 each
You lock one door — that never changes1/3
The other two doors, combined2/3
Monty is forced to open a goat from those twoalways
So that whole 2/3 lands on the one door he refused to open2/3

The reason Erdős wanted: Monty's door isn't a coin flip, it's a filter. He looked at both of the doors you didn't pick and deliberately removed a losing one. The door he leaves closed is the survivor of a search — yours is just the one you happened to touch first.

If three doors is too small to feel

Same game, 100 doors. Pick one, then watch Monty open 98 goats. Your door was a 1-in-100 guess and stays a 1-in-100 guess. The door he skipped past 98 times is the other 99%.

Your pick The door Monty left closed Opened, goat

Pick any door above — or just press the button and one gets picked for you.

Every round here is the real rules: the car is placed at random, your pick is random, and Monty always opens a door that is neither your pick nor the car. Read the whole story in the Monty Hall article.