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Monty Hall Problem
Three doors. One prize. Should you switch?
What is the Monty Hall Problem?
You're on a game show. There are three doors. Behind one is a prize. Behind the other two are goats. You pick a door. The host, who knows what's behind every door, opens a different door to reveal a goat. Then asks: do you want to switch to the remaining door, or stick with your original pick?
Why switching wins 2/3 of the time
Most people think it's 50/50 once a door is eliminated. It's not. Here's every possible scenario:
| Prize behind | You pick | Host shows | Switch wins? | Stick wins? |
|---|---|---|---|---|
| Door 1 | Door 1 | Door 2 or 3 | No | Yes |
| Door 1 | Door 2 | Door 3 | Yes | No |
| Door 1 | Door 3 | Door 2 | Yes | No |
No matter where the prize is, switching wins in 2 out of 3 scenarios. Your first pick has a 1/3 chance of being right. That never changes. The other two doors share the remaining 2/3 — and once the host eliminates one of them, all of that 2/3 probability concentrates on the one door left.
Still not convinced?
Play it yourself below, or use the Convince Me simulator to run hundreds of rounds and watch the 2/3 probability emerge in real time. Try it with 100 doors — that's when the intuition clicks.
Coop:Welcome to the Monty Hall Problem. Pick a door to get started.
Pick a door to begin
Your Stats
Switch win rate— (0 games)
Stick win rate— (0 games)
Play some rounds to see your stats!
Convince Me Mode
Watch the simulation play out. Try 100 doors to see why switching works.