In September 1990, a reader named Craig Whitaker sent a question to Marilyn vos Savant, who wrote the 'Ask Marilyn' column in Parade magazine and was famous for a stratospherically high recorded IQ. The question described a game show. You're given the choice of three doors: behind one is a car, behind the others, goats. You pick a door — say, No. 1. The host, who knows what's behind the doors, opens another door — say, No. 3 — revealing a goat. He then asks: do you want to switch to door No. 2? Is it to your advantage to switch?
Vos Savant answered: yes, switch. Sticking wins the car one time in three; switching wins two times in three. What followed was one of the great public meltdowns in the history of mathematics. Roughly ten thousand letters poured in, the overwhelming majority insisting she was wrong, including about a thousand from readers with PhDs — many of them mathematicians and statisticians. Some were withering. Surely, they argued, once one door is open, two doors remain, so it's fifty-fifty, and this famous genius was embarrassing herself in a national magazine.
She wasn't. The scenario is now known as the Monty Hall problem, after Monty Hall, the long-time host of the American game show Let's Make a Deal, and it has become the canonical example of how badly human intuition handles conditional probability.
Why switching wins
The cleanest way to see it: your first pick has a 1-in-3 chance of being the car. Nothing the host does changes that, because he will open a goat door no matter what you picked — he knows where the car is, and he never reveals it. So the remaining 2-in-3 probability doesn't evaporate; it collapses onto the single unopened door you didn't pick. Switch, and you win exactly when your first guess was wrong — which is two times out of three.
Still feels wrong? Scale it up. Imagine 100 doors. You pick one; the host, who knows where the car is, opens 98 other doors, all goats, leaving your door and one other. Do you really believe your initial 1-in-100 stab is now as good as the one door the host conspicuously left closed? The host's choice is not random — it's dripping with information.
The fine print matters, and it's where many disputes actually live. The 2/3 answer assumes the host always opens a door, always reveals a goat, and always offers the switch. If the host only offers a switch when you've picked the car, switching is a disaster. The problem as usually stated builds those rules in — and under those rules, the maths is airtight.
Erdős and the pigeons
The puzzle didn't start with vos Savant. A biostatistician named Steve Selvin had posed essentially the same problem in letters to the journal The American Statistician in 1975, coining the name 'Monty Hall problem.' And its skeleton is older still: it's a cousin of the Three Prisoners problem and of Bertrand's box paradox from the nineteenth century. But it was the Parade firestorm that made it famous — and demonstrated that credentials are no vaccine against it.
The most delicious casualty was Paul Erdős, one of the most prolific mathematicians of the twentieth century. When a colleague explained the problem to him, Erdős refused to accept the answer, and reportedly only came around after being shown a computer simulation demonstrating that switching wins about two-thirds of the time. If Erdős could get it wrong, everyone is allowed to.
There's an even more humbling footnote. In a 2010 study, psychologists Walter Herbranson and Julia Schroeder ran pigeons through a repeated version of the game with food rewards. The pigeons quickly learned to switch nearly every time. Human subjects given the same repeated trials adjusted far more slowly, clinging to their first choice. Birds, unburdened by the intuition that 'two doors means fifty-fifty,' simply followed the results.
Why is the wrong answer so magnetic? Psychologists point to a cluster of biases: we treat the two remaining doors as symmetrical when they aren't; we underweight the fact that the host's action is constrained by knowledge; and we suffer a special dread of switching away from a winner, which feels worse than losing by standing pat. Vos Savant eventually devoted several columns to the problem and suggested schoolchildren test it with cups and coins. Thousands of classrooms did. The experiments, unlike the angry letters, agreed with her.
The Monty Hall problem endures because it's a perfect trap: short enough to state in three sentences, simple enough to simulate at a kitchen table, and wrong-feeling enough that people will argue about it after every pub quiz where it appears.
Quiz nuggets
- In the Monty Hall problem, switching doors wins the car 2/3 of the time; sticking wins only 1/3.
- The problem is named after Monty Hall, host of the US game show Let's Make a Deal, and was first posed in this form by statistician Steve Selvin in 1975.
- Marilyn vos Savant's correct 1990 answer in Parade magazine drew around 10,000 letters of protest, including from roughly a thousand PhDs.
- Legendary mathematician Paul Erdős rejected the correct answer until he was shown a computer simulation.
- In a 2010 experiment, pigeons learned the optimal switching strategy faster than human subjects did.