The rules, on the table: three doors — behind one, a car; behind two, goats. You pick door 1. The host, who knows where the car is, opens door 3 — a goat bleats at you. He grins: 'Care to switch to door 2?'
Don't read on yet. Answer in your head: switch? Stay? Doesn't matter?
If your answer was 'two doors left, fifty-fifty, doesn't matter' — congratulations, you just fell into the same pit as nearly a thousand PhDs. The right answer: switch, and you win with probability 2/3; stay, and it's only 1/3.
The proof takes three lines: your first pick lands on the car with probability 1/3 — in that case switching loses for sure; your first pick lands on a goat with probability 2/3 — and then the host is forced to open the other goat door, so switching wins for sure. So 'switching wins' = 'your first pick was wrong' = 2/3. Statistician Steve Selvin posed the problem and gave this exact solution in a 1975 letter to The American Statistician — a full 15 years before the whole country started shouting about it.
⚠️The most spectacular collective faceplant on record: in September 1990, Marilyn vos Savant answered reader Craig Whitaker's three-door question in her Parade magazine column, arguing that 'switching doubles your odds to 2/3.' She then received about 10,000 letters, nearly 1,000 of them from PhDs — many of them mathematicians and statisticians; 92% of general readers' letters and 65% of letters from universities opposed her, all insisting 'two doors left, so 1/2 each.' The ending: on July 21, 1991, The New York Times confirmed on its front page that she was right; she also rallied classrooms across America to run the experiment, and the simulations uniformly backed switching at 2/3. The root of the error: everyone treated the host's door-opening as a random event carrying no information — but he knows where the car is and only ever opens a goat door; his choice is constrained by your first pick, and the 2/3 probability outside your first pick gets pressed entirely onto the one unopened door. Don't drop the premise: if the host doesn't know and just happens to reveal a goat, then and only then is it 1/2 either way.
Monty Hall: after the host opens a door, switching wins 2/3 of the time
