Drag to orbit · scroll or pinch to zoom · click a door to pick it (or press its number).
Where's the maths? This is probability, and specifically conditional probability: how the chance of something changes when you learn new information. Before the host moves, every door is 1/3. His move is not random noise — he knows where the car is and must avoid it — so opening a door is information, and the rules for updating probabilities when information arrives are exactly what make switching worth 2/3. The simulator shows a second big idea, the law of large numbers: one game tells you almost nothing, but thousands of games pull the win rate onto the true probability.
Beyond the syllabus: conditional probability is not a named topic; here it is done by listing cases.
The car hides behind one door, goats behind the rest. Every door is equally likely, so your first pick is right 1 time in 3 and wrong 2 times in 3.
Monty is not guessing. He knows where the car is and always opens a door that is neither your door nor the car. That is why his choice carries information.
| Car is behind | Stay | Switch |
|---|---|---|
| Door 1 | WIN | lose |
| Door 2 | lose | WIN |
| Door 3 | lose | WIN |
Three equally likely cases, assuming you picked door 1.
Switching wins exactly when your first pick was wrong — so P(win by switching) = 2/3, and P(win by staying) = 1/3.
Why "50-50" feels right: after a goat is shown, two doors remain and they look symmetric. They are not. Your door was sealed at 1/3 before Monty acted; all the leftover probability, 2/3, is squeezed onto the one door he refused to open.
With n doors the host opens n − 2 goats, and switching wins (n − 1)/n. With 100 doors that is 99% — the 3-door case is just the hardest one to feel.
In 1990 Marilyn vos Savant gave the answer "switch" in Parade magazine and received about 10 000 letters telling her she was wrong, roughly a thousand of them from people with PhDs. She was right.
Try this: switch to 100 doors and play one round — watching Monty open 98 goats makes the 99% obvious. Then simulate 1 000 trials of both strategies and check the bars against the dashed 1/n and (n−1)/n lines.
Monty Fall option: if the host opens a door at random and it just happens to be a goat, the rounds where he hits the car are thrown away — and among what is left, staying and switching are both 1/2. The host's knowledge is doing all the work.