Monty Hall Simulator
Simulate the Monty Hall problem thousands of times, compare switching against staying, and see how the answer changes when the host behaves differently.
Last reviewed by the Radiatus Cloud team
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Switching wins two thirds of the time, and the reason is the host
Your first pick is right one time in three, and that never changes. When the host opens a door he knows is empty, he has not given you new information about your own door; he has concentrated all the remaining probability onto the single door he did not open. So the door you did not pick carries the other two thirds. The result feels wrong because it seems like two closed doors should be equally likely, and that intuition ignores that the host's choice was constrained by knowledge you do not have.
The host's rule is the entire problem
If the host opens a door at random and it happens to be empty, switching and staying are both fifty-fifty, because no knowledge was applied and nothing was concentrated. The famous answer depends on the host always knowing where the prize is, always opening an empty door, and always offering the switch. Change any of those and the answer changes, which is why the problem generated so much argument: people were answering different questions and each was right about their own.
More doors make it obvious
With a hundred doors, you pick one, and the host opens ninety-eight empty ones. Almost nobody wants to stay. Your original chance was one in a hundred, and the single remaining door now carries ninety-nine hundredths. It is exactly the same structure as the three-door version, and the only thing that changed is that the concentration is too large to argue with.
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Frequently Asked Questions
Why is switching better?
Your first pick is right one time in three and stays that way. The host, who knows where the prize is, concentrates the remaining two thirds onto the one door he did not open.
What if the host opens a door at random?
Then switching and staying are equal at fifty-fifty in the cases where the opened door happened to be empty. The famous answer depends entirely on the host knowing and always avoiding the prize.
Does it work with more doors?
Yes, and it becomes obvious. With a hundred doors and ninety-eight opened, your door holds one in a hundred and the other holds ninety-nine.
Why did so many people disagree?
Because they were answering different questions. The answer depends on the host’s rule, and the rule is usually left unstated, so each side was right about the version they had in mind.
Is this simulation actually random?
It uses the browser’s random number generator, and running it repeatedly gives slightly different results, which is what a simulation is for: the convergence is the evidence rather than any single run.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Choose the number of doors and run the simulation.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.