The Monty Hall Problem Explained: Why You Should Always Switch

Step by step

1. Three doors · one car · you pick door 1
   Three doors. One car. You pick door one.
2. The host knows · he opens a goat door · switch?
   The host knows where the car is. He opens another door: a goat. Now, do you switch?
3. Switch? Stay? Or 50-50?
   Switch, stay, or does it not matter? Take a second.
4. Most say 50-50 · let's check every case
   Most people say C: two doors left, so it's fifty-fifty. Let's check every case.
5. Car behind door 1: stay wins · switch loses
   Case one: the car is behind door one. The host opens a goat door. Stay, and you win. Switch, and you lose.
6. Car at 2: host opens 3 · switch wins
   Case two: the car is behind door two. The host has to open door three. Switch, and you win.
7. Car at 3: host opens 2 · switch wins
   Case three: the car is behind door three. The host has to open door two. Switch, and you win again.
8. Switch wins 2 of 3 cases → 2/3 · stay → 1/3
   Switching wins two cases out of three. Staying wins only one. So switch: two thirds, against one third.
9. First pick right 1 in 3 → switch wins the rest
   Why? Your first pick is right only one time in three. The host never opens the car door, so switching wins every time your first pick was wrong.
10. 100 doors · or test it: 3 cups, 1 coin
    Still not sure? Picture a hundred doors. You pick one. The host opens ninety-eight goats. Would you stay? Or test it yourself: three cups, one coin, twenty rounds each way.
11. 1990 · vos Savant: switch · ~10,000 letters disagreed
    In nineteen ninety, Marilyn vos Savant answered switch in Parade magazine. About ten thousand readers wrote in, nearly a thousand with PhDs, most saying she was wrong. She was right.
12. On a test: 3 cases · switch wins 2 → 2/3
    On a test: list the three equally likely places for the car. Switching wins in two of them. The probability of winning by switching is two thirds.