Monty Hall Problem Calculator

What Is the Monty Hall Problem?

The Monty Hall problem is a classic probability puzzle. Three doors hide one car and two goats. You pick a door. The host, who knows what is behind each door, opens a different door that always shows a goat. You may then stay with your original pick or switch to the remaining closed door. Related tools: Conditional Probability Calculator and Bayes Theorem Calculator.

Why Switching Wins More Often

Your first pick is correct with probability $1/3$. The other two doors together hold probability $2/3$. When the host opens a goat door, that entire $2/3$ concentrates on the remaining closed door. Switching therefore wins with probability $2/3$, while staying wins with probability $1/3$.

$$P(\text{win by stay}) = \frac{1}{3}, \quad P(\text{win by switch}) = \frac{2}{3}$$

How This Simulator Works

Choose how many games to run and whether to always switch or always stay. Each trial places the car at random, picks a door at random, opens a host goat door, then applies your strategy. The win rate should approach the theoretical value as the number of simulations grows.

Example Intuition

Imagine 300 games. About 100 times your first pick is the car; staying wins those, switching loses. About 200 times your first pick is a goat; the host reveals the other goat, so switching wins the car. Switching wins roughly 200 of 300 games.