Monty Hall Problem Calculator
Experience the famous probability paradox - play the game or run simulations
The Monty Hall Problem
Ready to Play?
You'll see three doors. Behind one is a car, behind the other two are goats. Choose a door, then decide whether to switch when given the chance!
Theoretical Probabilities
Key Insight: Switching gives you a 2/3 chance of winning!
Monty Hall Facts
Named after game show host Monty Hall
First popularized by Marilyn vos Savant in 1990
Counter-intuitive result confounded many mathematicians
Demonstrates importance of conditional probability
Understanding the Monty Hall Problem
The Setup
You're a contestant on a game show. There are three doors. Behind one door is a car (the prize you want). Behind the other two doors are goats. You pick a door, but before it's opened, the host opens one of the other doors, revealing a goat.
The Decision
The host then asks: "Do you want to switch to the remaining unopened door, or stick with your original choice?"
Why Switch?
- •Initially, your door has a 1/3 chance of having the car
- •The other two doors together have a 2/3 chance
- •When one is eliminated, the remaining door gets all 2/3 probability
Mathematical Proof
Using Conditional Probability
Let's say you chose Door 1, and the host opens Door 2:
- P(car behind Door 1) = 1/3
- P(car behind Door 3 | host opens Door 2) = 2/3
Alternative Explanation
Imagine 100 doors instead of 3. You pick one. The host opens 98 doors with goats, leaving one closed. Would you switch? Of course! The same logic applies to 3 doors.
Key Insight
The host's knowledge changes everything. By always opening a door with a goat, the host gives you information that increases the probability of the remaining door.
Historical Context
Marilyn vos Savant (1990)
When Marilyn vos Savant published the correct solution in Parade magazine, she received over 10,000 letters, including from PhDs, saying she was wrong. The controversy lasted years.
Experimental Verification
Multiple TV shows and experiments have verified the solution. The most famous was on "MythBusters," which confirmed switching wins about 2/3 of the time.