Overview
The Monty Hall problem is the most famous probability puzzle of all time, named after the host of the game show Let's Make a Deal. It became infamous in 1990 when thousands of readers — including mathematicians — insisted the counter-intuitive answer was wrong.
You pick one of three doors, the host opens a different door to reveal a goat, and then offers you the chance to switch. The surprising truth: switching wins two times out of three.
How to solve The Monty Hall Problem
- Your first pick has a 1-in-3 chance of hiding the car. That means there is a 2-in-3 chance the car is behind one of the other two doors.
- The host always opens a losing door — and crucially, they know where the car is, so this is not a random reveal.
- Opening a goat door doesn't change your door's 1-in-3 odds, but it collapses the whole 2-in-3 onto the single remaining unopened door.
- Therefore switching gives you the 2-in-3 door. Always switch.
The key insight
The host's knowledge is the whole game. Because they never open the car door, their choice leaks information — funnelling the 2/3 probability onto the door they left closed. If the host opened a door at random, switching wouldn't help.
Variations & echoes
- Scale it up: with 100 doors, you pick one, the host opens 98 goats, and switching wins 99% of the time — the intuition suddenly becomes obvious.
- It's a textbook case of conditional probability and Bayesian updating.
- Simulating thousands of games is the fastest way to convince a sceptic.
Frequently asked questions
Isn't it 50/50 once one door is open?
No. That would be true only if the host opened a door randomly. Because the host knowingly avoids the car, the two remaining doors are not equally likely — yours stays at 1/3.
How much does switching actually help?
It doubles your win rate, from 1/3 (staying) to 2/3 (switching).