You pick a door. The host, who knows where the prize is, opens another door to show a goat and offers you a switch. It feels like 50/50 now. It isn't. Check it yourself: on video, in a game, in a simulation, through five explanations, and with 20 doors.
The Monty Hall problem, also called the Monty Hall paradox, is named after Monty Hall, host of the American show Let's Make a Deal, which first aired in 1963. The show never ran this exact game, but the setting is the same: a host, doors, prizes, and the temptation to change your mind.
You're the finalist. In front of you are three closed doors. Behind one is a brand-new car, behind the other two are goats. The prizes were placed in advance, at random. Only the host knows what is where.
You point at one door, say door 1. It stays closed for now.
The host opens one of the two doors you didn't pick, say door 3. It's a goat. He never touches your door and never reveals the car. If both unpicked doors hide goats, he opens either one at random.
"Do you want to stay with door 1, or switch to door 2?" Two doors are closed, and one of them hides the car.
They open the door you settled on. If it's the car, you drive it home. If it's a goat, the goat is yours.
Should you switch? Or does it make no difference?
Most people say it makes no difference: it's 50/50 now. In 1990 Marilyn vos Savant wrote in Parade magazine that you should switch. She got around 10,000 letters saying she was wrong, nearly a thousand of them from people with PhDs. She was right.
Change any one rule and the answer changes. Round 4 shows how.
In the 2008 film 21, maths professor Micky Rosa (Kevin Spacey) puts a game-show puzzle to his student Ben Campbell (Jim Sturgess). Ben says he would switch doors and explains why.
21 (2008), dir. Robert Luketic · English · open on YouTube
What the film leaves out. The 2/3 answer holds only if the host always opens a goat door and always offers the switch. If he opened doors at random, or only offered a switch to players who had already picked the car, the answer would change. More on that in round 4.
Play 15 to 20 rounds and try both strategies. The panel on the right shows how often each one wins in your session.
The tick on each bar marks the theoretical result. A few games swing a lot, so look at round 3 too.
The computer plays for you: it picks a door at random, the host opens goat doors, then the player either stays or switches. The chart shows the win rate settling on the theoretical value.
Two closed doors do not mean 50/50. You chose your door at random. The other one was left closed by a host who knows where the car is and is not allowed to reveal it. The two doors are not equal. Here are five ways to see it; pick the one that clicks for you.
Say you picked door 1. The car is equally likely to be behind any door.
| Car behind | Host opens | Stay | Switch |
|---|---|---|---|
| #1 | #2 or #3 | car | goat |
| #2 | #3 | goat | car |
| #3 | #2 | goat | car |
| Wins | 1 of 3 | 2 of 3 | |
Switching loses only when your first pick was the car, and that happens in just one case out of three.
Imagine the host offered from the start: "Keep your door, or take both of the others." You would take both. Showing a goat changes nothing: at least one of those two doors always hides a goat, and the host just tells you which. The group's 2/3 chance now sits in a single door.
When the host knows where the car is, his choice is constrained, and that constraint carries information. When he opens at random, the goat could have turned up by chance, so it tells you less. Check it: the simulation throws out games where the host accidentally revealed the car.
The asymmetry is in the middle factor. If the car is behind your door, the host picks #2 or #3 at even odds. If the car is behind #2, the host has no choice: he must open #3. So seeing #3 opened fits "car behind #2" twice as well.
If three doors didn't convince you, scale it up. At twenty doors, intuition usually gives in.
20,000 simulated games with 6 doors.
While you hold your first door, its chance is frozen at 1/n: the host may not open it, so each goat he shows says nothing about it. The chance of every opened door flows into the other closed doors. When two doors are left, the whole (n−1)/n sits in one of them, and that is the moment to switch. Switch earlier and your new door freezes at a lower level, so you give part of the win away.
After Marilyn vos Savant's 1990 column in Parade, the magazine received about 10,000 letters, nearly a thousand of them from people with PhDs explaining that she was wrong. At the time she was listed in the Guinness Book of Records for the highest IQ.
Paul Erdős, author of more than 1,500 papers and one of the most prolific mathematicians ever, refused to accept the 2/3 answer. Only a computer simulation, shown to him by his colleague Andrew Vázsonyi, changed his mind. Your round 3 did the same thing.
In a 2010 study, pigeons and people played the Monty Hall game hundreds of times. The pigeons, which simply learned what earned food, came to switch almost every time. The humans never reached the best strategy: their 50/50 "logic" got in the way.
In a 1991 interview with The New York Times, Monty Hall showed the reporter that on the real show he never had to offer a switch. He could offer it to players who had already picked the car, and then switching would be a mistake. That's why the rule "the host always offers the switch" matters.
The same logic appears in Joseph Bertrand's box paradox (1889) and in the three prisoners problem that Martin Gardner published in 1959. The game show just gave an old idea doors and goats.
The film 21 is based on Ben Mezrich's book about MIT students who counted cards to beat Las Vegas casinos. The Monty Hall scene is a test of whether the hero can put calculation above gut feeling.
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