What is the Monty Hall problem? Why switching wins

The Monty Hall problem is a probability puzzle based on a TV game show. There are three doors: a car behind one, goats behind the other two. You pick a door. The host, who knows where the car is, opens another door to show a goat and offers you a switch. Switching wins the car two times in three; staying wins one time in three. Most people, including many mathematicians, first insist it's 50-50.

In short

• Your first pick is right one time in three, and nothing the host does changes that, so switching wins the other two times in three.
• The answer depends on the host's rules: he always opens a door you didn't pick, always shows a goat, and always offers the switch.
• The puzzle became famous in 1990, when a magazine columnist gave the right answer and thousands of readers wrote in to say she was wrong.

What exactly is the puzzle?

The puzzle is named after Monty Hall, the host of the American game show Let's Make a Deal. The statistician Steve Selvin set it out in two letters to the journal The American Statistician in 1975, the second of which gave it its name (Wikipedia).

The answer depends on rules that casual tellings often leave out, so here they are in full:

1. The car is placed behind one of the three doors at random.
2. You pick a door.
3. The host knows where the car is. He always opens one of the two doors you didn't pick, and it always has a goat. If both have goats, he picks one of them at random.
4. He always offers you the chance to switch to the last closed door.

Under those rules, you should switch.

Why does switching win two thirds of the time?

The shortest argument: when you first pick, you have a 1 in 3 chance of being right and a 2 in 3 chance of being wrong. The host's move can't change which door you picked or what is behind it, so staying still wins 1 time in 3. If you were wrong to start with, which happens 2 times in 3, the host has to open the only other goat door, and the remaining door must hide the car. Switching wins exactly when your first pick was wrong.

Proof 1: list every case

Say you always pick door 1. The car is equally likely to be behind each door.

Car is behind	Host opens	Stay with door 1	Switch
Door 1	Door 2 or 3 (at random)	Win	Lose
Door 2	Door 3 (forced)	Lose	Win
Door 3	Door 2 (forced)	Lose	Win

Each row is equally likely. Staying wins in 1 row of 3; switching wins in 2. Picking door 2 or door 3 first gives the same table, relabeled.

Proof 2: imagine 100 doors

Now there are 100 doors and one car. You pick door 37. The host, who knows where the car is, opens 98 of the other doors, all goats, leaving closed only your door 37 and door 84. Your door had a 1 in 100 chance when you picked it. The host carefully avoided the car while opening 98 doors, so door 84 carries the other 99 in 100. Few people would stay. Three doors are the same game, just with a smaller gap.

Proof 3: play it many times

Play it with three cups and a coin, or simulate it. In one computer run of 10,000 games under the rules above, staying won 3,332 times and switching won 6,668: almost exactly one third and two thirds. Running a simulation is what convinced Paul Erdős, one of the most prolific mathematicians of the 20th century, who had refused to accept the answer from argument alone (Utah State University).

Why does it feel like 50-50?

Because two doors are left, and two unknowns feel equally likely. The mistake is treating the host's door as random. It isn't: he knows where the car is and must avoid it, and he can never open your door. His choice carries information about the door he leaves closed, not about yours.

A second trap is thinking the game "resets" after a door opens, as if you were walking in fresh to two doors. Someone who arrived at that moment, with no idea which door you'd picked, would indeed win half the time by guessing. You aren't in that position: you know which door was protected from being opened and which wasn't.

Does the answer change if the rules change?

Yes, and that is the most useful lesson in the puzzle. The statistician Jeffrey Rosenthal works through the random-host version, "Monty Fall", and others (Rosenthal, 2008); the last two rows below follow from the same case-by-case reasoning.

Host's behavior	Chance switching wins
Knows where the car is, always opens a goat door, always offers a switch	2 in 3
Doesn't know, opens a door at random that happens to show a goat ("Monty Fall")	1 in 2
Offers the switch only when your first pick is right	0: switching always loses
Offers the switch only when your first pick is wrong	1: switching always wins

The ignorant host gives 50-50 because his reveal could have shown the car; that it didn't is less informative. On the real show, Monty Hall wasn't bound to offer a switch at all. Asked about it in 1991 by The New York Times, as quoted on Wikipedia, he said whether the host offers one "all depends on his mood." The famous answer is right for the puzzle as stated, and only for that.

What happened when Marilyn vos Savant answered it?

In September 1990 a reader, Craig Whitaker, sent the puzzle to Marilyn vos Savant's "Ask Marilyn" column in Parade magazine. She said to switch. About 10,000 readers wrote in, nearly 1,000 of them with PhDs, most saying she was wrong (Wikipedia), some in strong terms ("you are the goat," one wrote). She called on math classes across the country to test it with paper cups and pennies, and they reported that she was right (Rosenthal, 2008).

Still argued

Probabilists agree on the answer to the puzzle as stated above. What is argued is whether the 1990 column stated it fully. Its wording didn't say outright that the host must always open a goat door and always offer the switch, and some critics argued the confusion came partly from that gap. Defenders reply that a host who opens a door at random isn't how the game is usually understood. Both sides agree the answer changes with the host's rules.

The episode is a lesson in how confidently people trust their intuition about chance, and how credentials don't protect against it, a pattern the Dunning-Kruger effect is often invoked to describe. The cure is the one critical thinking recommends: state the rules, list the cases, and test the answer. Like survivorship bias, the puzzle turns on how information was selected before you saw it.

Puzzles like this also sit where mathematics meets philosophy: what does it mean to say a door "has" a 2 in 3 chance when the car is already behind one of them? The answer is that probability here measures what you know, and the host's move changes what you know.

Check yourself

• Without the table, explain why switching wins exactly when your first pick was wrong.
• Why does the 100-door version make the answer feel obvious?
• If the host doesn't know where the car is and opens a goat door by luck, what is the chance switching wins, and why?

Sources

• Monty Hall problem — Wikipedia
• Monty Hall, Monty Fall, Monty Crawl — Jeffrey S. Rosenthal, Math Horizons, 2008
• The Monty Hall Problem — History of Statistics, Utah State University