The Monty Hall Problem — interactive write-up

Here's the setup, from an old TV show. In front of you are three doors. Behind one is a car; behind the other two, goats. You point at a door. Then the host — who knows where the car is — opens one of the other two doors to reveal a goat, and asks a simple question: do you want to keep your door, or switch to the last unopened one?

Almost everyone's gut says it can't matter — two doors left, surely it's a coin flip. Play a few rounds and watch your own tally, then run a batch of a thousand games.

Pick a door. One hides a car, two hide goats.

When you stayed0/0

When you switched0/0

Don't trust me — simulate it

Run a batch and watch stay settle near 33% and switch near 67%.

Why switching wins

The trick is that the host's move isn't random — and it isn't innocent. When you first pointed at a door, you had a 1 in 3 chance of being right. That means there's a 2 in 3 chance the car is behind one of the other two doors. Opening a goat door doesn't change that; it just shows you which of those two to avoid.

So the whole 2-in-3 weight that was spread across the other doors collapses onto the single one the host left closed. Staying keeps your original 1-in-3. Switching claims the other 2-in-3.

The cleanest way to feel it: imagine a hundred doors. You pick one — a 1-in-100 shot. The host then throws open ninety-eight goat doors, leaving just yours and one other. Would you still call it a coin flip? Almost certainly the car is behind the door the host went out of their way to leave shut.

Another way to see it

There are only three ways the game can start, each equally likely:

Your first pick was the car (happens 1 time in 3) — switching loses. Your first pick was goat A (1 in 3) — the host must reveal goat B, so switching wins. Your first pick was goat B (1 in 3) — the host reveals goat A, so switching wins again. Switching wins in two of the three cases. That's the whole proof.

Why it fools almost everyone

When the column “Marilyn vos Savant” published the switch answer in 1990, thousands of readers wrote in to say she was wrong — including people with PhDs. Our intuition quietly assumes the host acts by chance, like flipping over a random door. But the host always reveals a goat, and never the door you chose. That hidden knowledge is exactly the information switching lets you cash in.

It's a small, sharp reminder that in probability the rules of the game matter as much as what you can see — and that the simplest-looking questions are sometimes the ones worth testing before you trust your gut. Which is why there's a simulate button above: don't take my word for it.