← BlogSep 1, 20263 min read

The Monty Hall problem: the three‑door game from 21, explained

I'm Heron, your host tonight. Before I learned this game, I spent a whole afternoon insisting that two closed doors have to be a coin flip.

They never were: one door sits at a third, the other at two thirds. It took me three drawings to see that, so let's get you there in one.

Why switching wins two doors out of three

Every possible game fits into three cases. Say you picked door 1; any door tells the same story, so the only thing left to vary is where the car sits.

Two cases out of three reward the switch. Staying wins only when your blind guess was already right: the 1/3 we started with.

The scene from 21, and the line Ben brushed off

The movie 21 (2008) runs this exact game in a lecture hall. Professor Micky Rosa puts Ben on a game show: three doors, a car, two goats, and “the game show host, who, by the way, knows what's behind all the other doors,” in the movie's own words.

Ben picks door 1, the host opens door 3 and shows a goat, and Ben switches: 33.3% for his first pick, 66.7% for the other door. He calls it “variable change,” and the numbers are right.

Then the professor asks the best question in the scene: “How do you know he's not playing a trick on you?” Ben answers, “Well, I wouldn't really care. I mean, my answer's based on statistics,” and that shrug is the line to care about.

The numbers are right, the shrug is wrong: a host who offers the switch only when you already sit on the car makes switching lose every single time, a variant known as “Monty from Hell”. The 66.7% holds only when the host opens a goat every game, whatever you picked.

The coin flip that never happened

Two closed doors feel like even money, and that feeling counts doors. Doors were never what carried the odds we're counting.

Your door and the survivor were chosen in different ways. You grabbed yours blind; the host picked his knowing exactly where the car was.

We're calling this one The Frozen Pick: the second you touched your door, its odds froze at 1 in 3, and no later reveal thaws them.

This coin-flip feeling has a track record. After Marilyn vos Savant answered “switch” in Parade in 1990, about 10,000 readers wrote to the magazine, most of them telling her she was wrong, per The New York Times, 1991.

In a 1995 experiment with 228 people, only 13% switched. Compare that with what readers pick in the game above.

My mistake was re-picking my first door, as though I were choosing fresh between two doors.

You never are: that door was decided while three doors were closed.

Run it with 100 doors

Same game, and now let's make the board enormous. You pick 1 door out of 100, the host opens 98 goats, and two doors are left.

Your door is still the blind grab; the other survived 98 rounds of a host who never opens the car. Type the percentage of games where switching wins.

Same trap, louder. The missing dollar in our math riddles list is the same species.

When it really is a coin flip

One condition holds our whole answer up: the host has to know where the car is. He opens a goat on purpose, every single time.

The professor in 21 says the host “decides to open another door,” and everything hangs on whether he does that every game.

Swap him for someone opening a leftover door at random and our math changes: that host sometimes reveals the car, and switching and staying each win half of the games that survive.

What to remember

• Your first pick freezes. One guess out of three, locked in before anything opened.
• Count the shares, then the doors. The two doors you skipped held 2/3, and the host parked all of it on one.
• Ask who knew what. A host with knowledge hands you 2/3; a guessing host hands you half.

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