Online Tool Store
Monty Hall Simulator
Simulate the Monty Hall problem over thousands of games and watch the switching win rate converge on two-thirds while staying stays at one-third.
🔒 This tool runs entirely in your browser. Your files are never uploaded to a server.
Monty Hall Simulator
Play a round, then run thousands of games to see why switching wins.
- 1 Pick
- 2 Host reveals
- 3 Stay / switch
- 4 Result
Run a simulation
Theory: staying wins 1/3, switching wins 2/3.
How the Monty Hall Simulator works
- Pick a strategy, or compare both at once.
- Run enough games that random noise settles — a hundred games can easily look like a coin flip.
- Try more doors: with ten doors and eight opened, switching wins nine times out of ten, which makes the intuition obvious.
The method
Your first pick is right one time in three, and the host's knowledge means the remaining door carries all the probability your original pick did not.
P(win | switch) = 1 - 1/n = 2/3 for three doors; P(win | stay) = 1/3
With ten doors the effect is stark: your first pick is right 10% of the time, so switching wins 90% — the same logic, just harder to argue with.
FAQ
Why is it not fifty-fifty?
Because the host does not open a door at random — he knows where the prize is and never reveals it. That knowledge transfers information to the door he leaves closed.
What if the host opens a door randomly?
Then the games where he happens to reveal the prize are discarded, and among the rest switching and staying genuinely are equal. The host's knowledge is the entire mechanism.
Why do so many people get this wrong?
Because the intuition treats two closed doors as symmetric. They are not: one was chosen by you at random, the other survived a deliberate filter.
Does the switching advantage hold with more than three doors?
Yes, and it gets stronger. Run the simulator with ten doors and eight opened: your first pick is only right 10% of the time, so switching wins about 90% of games — the same logic as three doors, just harder to argue with.
How many simulated games do I need before the win rate settles?
A few hundred games is usually enough to see switching pull clearly ahead of staying; a hundred or fewer can still look close to a coin flip just from random noise.
How we compare
| Feature | Online Tool Store | A graphing calculator | A stats package |
|---|---|---|---|
| Runs thousands of trials | ✓ | ✗ | ✓ |
| Variable door count | ✓ | ✗ | Sometimes |
| Shows both strategies | ✓ | Slowly | ✓ |
| No install | ✓ | ✓ | ✗ |
Monty Hall Simulator settles the argument empirically, and the ten-door setting is what usually converts the last sceptic in the room.