Table of Contents
The Monty Hall Problem
The Monty Hall problem is a famous probability puzzle based on the TV game show "Let's Make a Deal." You choose one of three doors. Behind one door is a car; behind the others, goats. The host (who knows what is behind each door) opens another door showing a goat, then asks if you want to switch your choice.
Counter-intuitively, you should always switch! Switching gives you a 2/3 chance of winning, while staying gives only 1/3. This result has confused many people, including mathematicians, since it was popularized by Marilyn vos Savant in 1990.
The Solution
For the classic 3-door problem: staying wins 1/3 of the time, switching wins 2/3 of the time. The key insight is that the host's action of revealing a goat provides information that changes the probabilities.
The Mathematics
| Doors | P(Stay Wins) | P(Switch Wins) |
|---|---|---|
| 3 | 33.33% | 66.67% |
| 4 | 25.00% | 75.00% |
| 5 | 20.00% | 80.00% |
| 10 | 10.00% | 90.00% |
Frequently Asked Questions
Why is switching better?
When you first choose, you have a 1/3 chance of being right. That means there is a 2/3 chance the car is behind one of the other doors. When the host reveals a goat, all that 2/3 probability concentrates on the remaining unchosen door.
Does this work with more doors?
Yes, and the advantage of switching increases with more doors. With 100 doors, your initial pick has a 1% chance. After 98 goat doors are revealed, the remaining door has a 99% chance. Always switch!
What if the host opens a door randomly?
If the host does not know what is behind the doors and opens one randomly (sometimes revealing the car), then switching and staying have equal probability. The host's knowledge is crucial to the problem.