2 3 Monty Hall

You are on a game show. In front of you are three closed doors. Behind one is a brand-new car. Behind the other two: goats.

You point at Door 1.

The host, who knows exactly where the car is, walks over and opens Door 3. A goat blinks back at you.

"Would you like to switch to Door 2?" he asks.

Most people shrug. Two doors left, one car. Surely it's fifty-fifty?

It isn't. If you switch, you win the car two times out of three.

In 1990, saying so in print caused one of the loudest arguments in the history of popular mathematics.

Retro 1970s television game-show stage: three numbered doors under spotlights, a host in a wide-lapelled suit gesturing at open Door 3 with a goat, a contestant hesitating in the foreground

September 1990

A letter to "Ask Marilyn"

In September 1990, a reader named Craig F. Whitaker sent a puzzle to Marilyn vos Savant, who wrote the "Ask Marilyn" column in Parade, a Sunday magazine tucked into newspapers all over the United States.

Vos Savant was listed in the Guinness Book of World Records Hall of Fame for "Highest I.Q." - a fact her readers were reminded of every week.

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to take the switch?"

Her answer was short and confident: yes, you should switch. Switching, she said, wins two times in three.

She could not have guessed what the postman was about to bring.

A Sunday newspaper spread on a kitchen table, a magazine insert open at a question-and-answer column, a cup of coffee, reading glasses, and a small margin doodle of three doors and a goat

The Postbag

Ten thousand letters

By her own estimate, vos Savant received about 10,000 letters. The great majority said she was wrong.

Of the critical letters, close to 1,000 were signed by people with PhDs, many on the letterheads of university mathematics and science departments.

"You blew it!" wrote Robert Sachs, a professor of mathematics at George Mason University. "As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful."

"Our math department had a good, self-righteous laugh at your expense," wrote another professor.

"You are utterly incorrect," insisted a professor at Georgetown University. "How many irate mathematicians are needed to get you to change your mind?"

One letter simply told her: "You are the goat!"

Vos Savant did not budge. She wrote more columns explaining herself - and invited schoolchildren to test the game in class. Teachers at more than 1,000 schools, from second grade to graduate level, tried it. The results kept coming back the same way.

A mountain of envelopes spilling from mail sacks across a desk, some stamped with university crests, a few with red ink exclamation marks, a calm figure sorting them with a knowing smile

Margin note

Humble pie.

Most of the critics never wrote back to admit they had been wrong. Robert Sachs did. "I wrote her another letter," he told The New York Times in 1991, "telling her that after removing my foot from my mouth I'm now eating humble pie."

The Logic

Why switching wins

Here is the secret: the host is not guessing. He knows where the car is, and he will always open a door with a goat.

When you first point at a door, you have a 1 in 3 chance of being right. Nothing the host does afterwards changes that. He was always going to show you a goat, whatever you picked.

So there are only three ways the game can go:

1
You picked the car. The host opens either goat door. Switching loses.
2
You picked Goat A. The host must open Goat B's door. Switching wins.
3
You picked Goat B. The host must open Goat A's door. Switching wins.

Stick, and you win in one of the three. Switch, and you win in two of the three.

Stick: 1/3. Switch: 2/3.

If it still feels wrong, imagine 100 doors. You pick one. The host, who knows where the car is, opens 98 others - all goats - and leaves just one closed. Would you really stay with your first wild guess?

The two remaining doors look equal. They are not. One was chosen by you, at random. The other was left standing by someone who knew.

Comic-strip style diagram showing three rows for each starting pick, the host opening a goat door in each, and switch winning two out of three rows

Three playing cards on a kitchen table, one flipped to show a two of clubs, with a tally sheet beside them

Try it yourself

Try it at home.

You don't need a game show. Use three playing cards: one ace, two others. A friend shuffles them face down and plays the host, always turning over a non-ace after you choose. Play twenty rounds sticking and twenty rounds switching. Keep score. The switching pile will usually be about twice as big.

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1995 · Paul Erdős

The genius who wanted a reason

If you are still not convinced, you are in excellent company.

In 1995, the mathematician Andrew Vázsonyi told the puzzle to his house guest, Paul Erdős - one of the most prolific mathematicians who ever lived.

Erdős's reply, as Vázsonyi remembered it:

"No, that is impossible, it should make no difference."

Vázsonyi showed him the logic. It did not help. Erdős wanted a plain, common-sense reason.

So Vázsonyi tried something else. He ran a computer simulation - the game played over and over, 100,000 times, keeping a running score. Switching won about two times in three.

Erdős was reluctantly convinced. But he still said he did not understand why.

A few days later, Vázsonyi wrote, Erdős phoned to say that a friend had finally explained the reasoning to him. The friend was Ron Graham - the same mathematician you meet on our /grahams-number page.

An elderly bespectacled Erdős in a rumpled suit with headphones around his neck, frowning at a chunky 1990s computer screen showing a running tally of cars and goats, Vázsonyi beside him with arms folded

July 1991 · Beverly Hills

Monty Hall plays his own game

The puzzle is named after Monty Hall, the real host of the American game show Let's Make a Deal, which he hosted for more than 4,500 episodes.

In July 1991, a New York Times reporter went to Hall's home in Beverly Hills to test it on the man himself.

Hall set up three miniature cardboard doors on his dining-room table. An ignition key played the car. The goats were played by a packet of raisins and a roll of Life Savers.

In ten rounds where the reporter stuck with his first door, he won 4 cars. In ten rounds where he switched, he won 8.

"So her answer's right: you should switch," Hall said.

Then he grinned, and pointed out a loophole. On his real show, he was never required to open a door or offer a swap. A host who only offers a switch when you have already picked the car can make switching a disaster.

"If the host is required to open a door all the time and offer you a switch, then you should take the switch," he said. "But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood."

A smiling silver-haired game-show host at a dining-room table with three little cardboard doors, a car key, a box of raisins and a roll of mints, a reporter with a notebook keeping score

A 1970s academic journal lying open beside a vintage magazine, both showing tiny drawings of three doors

Margin note

Older than the column.

The puzzle did not start with Parade. A statistician named Steve Selvin described it in letters to the journal The American Statistician in 1975 - and his second letter was the first time anyone called it "the Monty Hall problem." An earlier cousin, the "three prisoners" puzzle, appeared in Martin Gardner's Scientific American column in 1959. Gardner warned that "in no other branch of mathematics is it so easy for experts to blunder as in probability theory."

The Bigger Idea

Our brains weren't built for this

Why do so many clever people get it wrong?

Because our instincts see two closed doors and think "one each". They forget to ask who opened the third door, and why.

The host's knowledge is information. When he skips over a door, that tells you something. Switching lets you take advantage of everything he knows.

"Our brains are just not wired to do probability problems very well," the Harvard statistician and former magician Persi Diaconis told the Times, "so I'm not surprised there were mistakes."

Probability is full of these traps. On our /23 page, a roomful of strangers fools almost everyone in a different way.

The lesson of 2/3 is not "trust the experts" or "distrust the experts". It is: when intuition and careful counting disagree, count again - or play the game a hundred times and see.

Two tall closed doors in a quiet landscape, a faint trail of footprints leading away from a third open door, a lantern casting more light over one closed door

The Exhibit Label

2/3 in a nutshell

Here is why 2/3 matters:

In the Monty Hall game, you pick one of three doors, the host (who knows where the car is) opens a goat door, and offers you a swap.
Sticking wins 1 time in 3. Switching wins 2 times in 3.
In 1990, Marilyn vos Savant said so in Parade and received about 10,000 letters, the great majority insisting she was wrong - close to 1,000 of the critics had PhDs.
Schoolchildren all over America tested it and found she was right.
Paul Erdős was only reluctantly convinced by a computer simulation.
Monty Hall himself agreed - as long as the host must always open a door and offer the switch.

Final recap illustration combining three doors with a goat, a mountain of letters, Erdős at a computer, Monty Hall's dining-room table with raisins and mints, a glowing 2/3 at the centre

The odds that made a thousand PhDs write angry letters