The full story
This is the episode's narration, word for word. Headings jump to that point in the video.
Three doors 0:00
You're on a game show, facing three closed doors. Behind one is a car, and behind the other two, goats. You pick door one. The host, who knows where the car is, opens door three: a goat. Do you want to switch to door two? Two doors, one car, so surely it's fifty-fifty. That's what most people think, and it's wrong. Switch, and you win two times in three. So where does the extra chance come from?
The fine print 0:34
First, the exact rules, because the whole puzzle lives in them. The car is placed at random, and the host knows where it is. He always opens a door you didn't pick, always shows a goat, and always offers the switch. If your first pick was the car, he chooses between the two goats at random. Change a rule, and the answer can change.
Boxes and car keys 0:59
In February 1975, a journal called The American Statistician printed a short puzzle from Steve Selvin, a biostatistician at the University of California, Berkeley. It used the TV show Let's Make a Deal: three boxes, car keys in one. It wasn't a new idea: the puzzle writer Martin Gardner had written about a cousin of it, the three prisoners problem, in 1959. Readers wrote in to say he was wrong. In August he replied, spelling out the rule his answer rested on: the host knows where the keys are, and when either of two boxes is empty, he picks one at random. He titled that reply On the Monty Hall Problem.
Play every game 1:47
So let's play every game. You pick door one. If the car's behind door one, the host opens either goat, and switching loses. If it's behind door two, his only goat is door three, so you switch to two and win. If it's behind door three, he must open two, so you switch to three and win again. Switching loses only when your first pick was right, which happens one time in three. So switching wins the other two.
Ask Marilyn 2:20
On the ninth of September 1990, Marilyn vos Savant, who wrote Ask Marilyn, a column in the American magazine Parade, printed the puzzle, sent in by a reader named Craig Whitaker. Her reply was blunt: yes, you should switch. The first door has a one in three chance, she wrote, and the second door two in three. To make it obvious, she asked readers to picture a million doors.
A hundred doors 2:50
A hundred doors will do. You pick one, so your chance is one in a hundred. The host, who knows where the car is, opens ninety-eight of the others, all goats, and leaves one door shut. He could do that whatever you picked, so it tells you nothing about your own door. But he had to steer around the car, so the one door he left shut carries the chance of all ninety-nine you didn't pick. Three doors work the same way, just smaller.
You are the goat 3:20
Her readers were not persuaded. By her count, ninety-two per cent of letters from the public said she was wrong, and sixty-five per cent of those from universities. By her own estimate, reported in the New York Times, about ten thousand letters arrived, and close to a thousand of the critical ones were signed by people with PhDs. A mathematician at George Mason University wrote: you blew it. One from Georgetown University asked how many irate mathematicians it would take to change her mind. Another simply wrote: you are the goat.
Pennies and paper cups 3:59
So she set the country some homework. Maths classes were to play the game with three paper cups and a penny, two hundred times staying and two hundred times switching. A reader at Los Alamos National Laboratory wrote that two of his colleagues had each programmed it, and in a million trials, switching paid off 66.7 per cent of the time. And the George Mason mathematician wrote again, to say he was eating humble pie.
Run it yourself 4:29
You can check it yourself. We had a computer play a million games, with the car placed at random and the host following every rule. The longer it ran, the closer the score settled: switching wins about two games in three, and staying about one. That's not a trick of wording; it's what happens when you play.
The man who loved only numbers 4:51
Paul Erdős was perhaps the most prolific mathematician of the twentieth century. In 1995, by his friend Andrew Vázsonyi's account, Erdős heard the answer and said no, that is impossible, it should make no difference. A decision tree didn't move him. A computer simulation did, but he complained that it didn't tell him why. Later, his colleague Ron Graham explained it, and then he understood.
What the host knows 5:22
Here's why. The host's choice is evidence, and you weigh evidence by asking how likely it was in each possible world. You picked door one. If the car's behind door one, the host could open either other door, so he opens three half the time. If it's behind door two, he has to open three. If it's behind door three, he never opens it. So what you saw is twice as likely in the door-two world as in the door-one world.
Weighing the worlds 5:53
That makes the door-two world twice as likely as the door-one world, and the door-three world is gone. Two parts to one: two in three for switching, one in three for staying. Updating a chance after you learn something is called conditional probability, and the rule for the weighing is Bayes' theorem. Start with what you believed, multiply by how well each possibility predicts what you saw, then rescale so it adds up to one.
Monty plays Monty 6:24
In July 1991, a New York Times reporter took the puzzle to Monty Hall himself, host of Let's Make a Deal. At his dining table, with cardboard doors and goats played by raisins and a roll of sweets, they played twenty games. Staying won four cars in ten, and switching won eight. Then Monty changed the game. When the reporter picked a goat, Monty just opened that door; when he picked the car, Monty offered the switch. He won the next eight rounds.
The host who slips 7:00
So the answer depends on the host. Suppose he doesn't know where the car is. He slips on a banana peel, knocks open one of the other two doors at random, and it happens to show a goat. That goat was just as likely in the door-one world as in the door-two world, so now it really is fifty-fifty. Our simulated clumsy host agrees: when his fall revealed a goat, switching won half the time. Same doors, same goat, different answer, because this open door carries no knowledge.
Lazy hosts and devious ones 7:36
Say the host is lazy, and when he has a choice he always opens the lowest-numbered door. If you pick door one and he opens door three, he was forced to, so the car must be behind door two. If he opens door two, it's back to fifty-fifty. And if he offers the switch only when you've already got the car, switching never wins. Martin Gardner said the problem isn't well-formed unless the host must always open an empty door and offer the switch.
Two doors, so fifty-fifty 8:07
So why does most people's gut say fifty-fifty? In a 1995 study by Donald Granberg and Thad Brown, only about one person in eight switched the first time. With practice, that rose to about half. One name for the hunch is the uniformity belief: with two unknowns left, we split the chance evenly, whatever their history. It ignores the one thing that matters, which is that the host knew where the car was.
The fear of switching 8:39
There's a second force, and it's regret. Granberg and Brown found people would feel more frustrated and angry if they switched and lost than if they stayed and lost. Monty Hall knew that from the studio floor. Contestants thought their chance had gone up to one in two, he said, so they hated to give up their door, no matter how much money he offered.
The pigeons 9:03
Then there are the pigeons. In 2010, Walter Herbranson and Julia Schroeder, at Whitman College in Washington State, gave the game to six pigeons, with three lit keys as doors and a little grain as the prize. On day one the birds switched just over a third of the time; by day thirty, ninety-six per cent. A small group of students played the same game, with no story. After two hundred rounds, they were still switching only about two times in three.
Stand behind the doors 9:37
What does help is changing where you stand. A 2003 study found people did much better when the puzzle was told from the host's side, and in counts rather than chances. So stand behind the doors and watch thirty games. In about ten, the car is behind your door, and switching loses. In the other twenty, the host's hands are tied: he must open the only other goat, so the door he leaves shut hides the car, and that's about twenty wins in thirty.
So, should you switch? 10:10
So should you switch? If the host knows where the car is and plays by the rules, then yes: switching doubles your chance, from one in three to two in three. If he's guessing and just happens to show a goat, it makes no difference. If he chooses when to offer it, beware: as Monty Hall put it, it all depends on his mood. The doors never change; what changes is what the host's choice tells you. Next time, another chance that feels wrong but isn't: why do twenty-three people probably share a birthday?
Sources
Every factual claim in the episode is tied to one of these. Spotted an error? Tell us.
- Marilyn vos Savant, "Game Show Problem" (the Ask Marilyn columns and letters as… — web.archive.org
- Donald Granberg and Thad A. Brown, "The Monty Hall Dilemma", Personality and Social… — doi.org
- Stefan Krauss and X. T. Wang, "The Psychology of the Monty Hall Problem: Discovering… — web.archive.org
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd rev. ed., American… — math.dartmouth.edu
- Jeffrey S. Rosenthal, "Monty Hall, Monty Fall, Monty Crawl", Math Horizons, September… — probability.ca
- John Tierney, "Behind Monty Hall's Doors: Puzzle, Debate and Answer?", The New York… — nytimes.com
- Steve Selvin, "On the Monty Hall Problem" (letter to the editor), The American… — web.archive.org
- Steve Selvin, "A Problem in Probability" (letter to the editor), The American… — doi.org
- Monty Hall, letter to Steve Selvin, 12 May 1975, reproduced on "The Monty Hall… — web.archive.org
- Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search… — archive.org
- Andrew Vazsonyi, "Which Door Has the Cadillac?", Decision Line (Decision Sciences… — web.archive.org
- Walter T. Herbranson and Julia Schroeder, "Are Birds Smarter Than Mathematicians?… — pmc.ncbi.nlm.nih.gov
Researched and scripted with AI assistance, fact-checked claim by claim, with synthetic narration and diagrams drawn in code. How we make episodes.
