Every Monty Hall game, played at once

2026-07-07

Three doors. Behind one, a car; behind the other two, goats. You pick a door. The host - who knows exactly where the car is - opens one of the doors you didn't pick, always revealing a goat, and offers you a deal: stick with your door, or switch to the last unopened one.

Two doors left, one car. It feels like a coin flip - 50/50, switching can't matter. That feeling is wrong, and the fastest way to see why is to stop playing one game and play all of them at the same time.

Each little cell is one universe. In every single one you picked door 1 - the outlined door on the left. The only thing that varies from universe to universe is where the car landed: the yellow dot. The crossed-out door is the goat the host opened for you. Now flip stay / switch and watch the wall invert. Green means your strategy wins that universe. Press the reroll button for a fresh set of worlds; the ratio barely moves.

Count the worlds

Seen all at once, the probability question becomes a counting question. Staying wins in exactly the universes where your first pick was already the car. The car was placed before you chose, with no idea what you would choose - so it sits behind door 1 in one universe out of three. Nothing the host does afterwards touches your door or moves the car. Stay wins 1/3 of the wall, and no amount of door-opening theatre can change which third.

And in every universe, exactly one of the two strategies wins: if staying loses, the car is behind the other closed door, so switching wins. The green cells under stay and the green cells under switch are perfect complements - the same wall, inverted. So switch wins the other 2/3. Not sometimes. Two-thirds of all possible worlds.

What the host actually does

Look closely at the universes where your first pick was wrong - two out of every three. In those worlds the host has no freedom at all. One of the doors you didn't pick hides the car; he can't open that one. He is forced to open the only goat left, and the remaining closed door must hide the car. His reveal doesn't split the odds between two doors - it funnels the whole 2/3 onto one of them.

That is what "the host gives you information" means, made literal. Switching isn't a hunch about where the car is. It's a bet that your first guess was wrong - and first guesses are wrong two times out of three.

The letters

In 1990 Marilyn vos Savant published this answer - switch, it's 2/3 - in Parade magazine, and around ten thousand readers wrote in to tell her she was wrong. Roughly a thousand of the letters came from PhDs, some of them mathematicians, many of them insisting the answer was 50/50. Her best reply was essentially the diagram above: don't argue about one game, play out all the cases and count. When schools and labs ran the simulation, the switchers won two-thirds of the time, and the letters stopped.

One game feels like a paradox. A hundred and eighty games are just a census: your first pick was right in a third of the worlds, and the host quietly hands you the rest. When probability confuses you, build the wall of universes and count the green.