Mathematics
What is The Monty Hall problem?
The Monty Hall problem is a famous probability puzzle: behind three doors are two goats and one car; after you pick, the host opens a goat door and offers a switch. Counterintuitively, switching wins the car two-thirds of the time, not half.
See it, don’t just read it.
Watch a 2-minute lesson with voice + animation that explains the monty hall problem.
Key things to understand
- 1Your first pick has a 1/3 chance of the car; the other two doors together hold 2/3.
- 2The host knowingly opens a goat, concentrating that 2/3 onto the one remaining unopened door.
- 3Switching therefore wins 2/3 of the time; staying wins only 1/3.
- 4It fools almost everyone because the host's knowledge changes the odds — it isn't a fresh 50/50.
Frequently asked questions
- Why isn't it just 50/50 after a door opens?
- Because the host never opens the car or your door — that knowledge transfers the eliminated door's probability to the other door, not to yours.
- Does switching guarantee a win?
- No. Switching wins about 2 of 3 times; you can still lose a single game, but over many games switching is clearly better.
- Who is it named after?
- Monty Hall, host of the game show 'Let's Make a Deal,' whose door-reveal format inspired the puzzle.