Should You
Switch Doors?
Three doors. One car. Two goats. Pick a door, watch the host reveal a goat, then decide: stay, or switch. Play a few rounds and watch the odds add up.
Pick a door.
Your results so far
Dashed lines mark the true odds: 1 in 3 for staying, 2 in 3 for switching.
Act two
Still not convinced? Try 100 doors.
The reasoning is exactly the same whether there are three doors or three hundred. It just feels more obvious once there are enough of them that "the host knows something you don't" becomes impossible to ignore.
You pick one door. The host, who knows what's behind every door, opens every other door except one, all goats. Only your door and one other are left closed.
The math behind the door
In 1990, a reader sent this exact puzzle to Marilyn vos Savant's "Ask Marilyn" column in Parade magazine. She answered correctly: switch. What followed was one of the great episodes in popular math history. Thousands of readers wrote back to say she was wrong, including hundreds who identified themselves as having a PhD. Several were mathematicians. She had not made a mistake. They had.
When you first pick a door, you have a 1 in 3 chance of picking the car. That means there's a 2 in 3 chance the car is behind one of the other two doors.
The host knows what's behind every door and always opens a goat. That doesn't change your original 1 in 3 chance: it just concentrates the remaining 2 in 3 chance onto the one door you didn't pick and the host didn't open.
| If the car is behind | You picked | Host opens | Stay | Switch |
|---|---|---|---|---|
| Door 1 | Door 1 | Door 2 or 3 | Win | Lose |
| Door 2 | Door 1 | Door 3 | Lose | Win |
| Door 3 | Door 1 | Door 2 | Lose | Win |
Assuming you always pick Door 1 first (by symmetry, the logic is identical for any starting door), switching wins in 2 of the 3 possible positions of the car. Staying wins in only 1.
The same formula scales to any number of doors: with N doors, staying wins 1 in N of the time, and switching wins the remaining N-1 in N. That's why the 100-door version above feels so lopsided: your first guess still only had a 1% chance of being right, and the host revealing 98 goats never touched that.