Why Switching Doors Wins Twice as Often

In September 1990, Marilyn vos Savant answered a reader's letter in Parade magazine. The setup: three doors, one hides a car, two hide goats. You pick door 1. The host, who knows what's behind each door, opens door 3 to show a goat. He asks if you want to switch to door 2. Should you?

Vos Savant said yes — switching wins two times out of three. The column drew about 10,000 letters disagreeing, including from roughly 1,000 readers with PhDs. She was right.

The key is that the host's reveal is not random. When you first pick, you have a 1/3 chance of being on the car. That 2/3 probability of being wrong does not evaporate when a goat is shown — it concentrates on the one remaining unopened door. So the unopened door you didn't pick now carries 2/3 of the probability mass, while your original pick still carries 1/3.

A cleaner way to feel the asymmetry: imagine 100 doors. You pick one (1% chance of car). The host opens 98 goat doors. Would you switch to the one door he left closed? Almost certainly yes — that door is now carrying 99% of the probability.

The puzzle is named for Monty Hall, host of Let's Make a Deal from 1963 to 1986. The actual game show never quite ran the puzzle as stated; mathematicians cleaned it up afterward into the version that broke the internet thirty years before the internet was ready to be broken.