How to use the Monty Hall simulator
A car sits behind one of three doors. You pick one, and the host opens another door to reveal a goat. Should you switch? Intuition says it is now fifty-fifty, but switching is twice as good. This simulator settles the argument by playing the game tens of thousands of times in front of you. Press the button and it randomises the car and your first pick, then walks through the host opening doors, trial after trial.
The theoretical values sit next to the simulated ones for a reason. Switching wins exactly when your first pick was wrong, which is (n − 1) ÷ n for n doors, while staying wins when the first pick was right, or 1 ÷ n. With three doors that is two thirds against one third, about 66.67% versus 33.33%. Raise the trial count from a hundred to a hundred thousand and you can watch the simulation converge onto those numbers.
More doors make the effect starker. With ten doors the host opens eight of them, so switching wins 90% of the time. Doors can run from 3 to 10; with only two there is no goat door left for the host to open and the puzzle falls apart. Because the simulation is random, small changes between runs are expected.
Frequently asked questions
Your first pick is wrong two times out of three, and whenever it is wrong the door left after the host opens a goat door must hide the car. So switching wins two thirds of the time.
No, that is sampling noise, and it shrinks as the number of trials grows. The gap is reported in percentage points alongside the rates.
The host opens more doors, so switching wins (doors − 1) ÷ doors. With ten doors that is 90%.