Monty Hall Problem Calculator
See the real winning probabilities behind the classic Monty Hall puzzle: should you stay with your original door or switch after the host reveals a goat?
What the Monty Hall problem is
The Monty Hall problem is a famous probability puzzle named after the host of the game show “Let’s Make a Deal.” You pick one of several doors, one of which hides a prize. The host, who knows what’s behind every door, then opens one or more of the remaining doors that do not hide the prize. You are offered the chance to switch your choice to another unopened door. Counterintuitively, switching usually gives you better odds than staying.
The formula
With n total doors and r non-prize doors revealed by the host, your original door still has a win probability of 1/n if you stay. If you switch, the remaining probability mass (1 – 1/n) is spread evenly across the doors still closed other than yours, giving a switch win probability of (n-1) / (n × (n-1-r)). In the classic 3-door, 1-reveal version this simplifies to the well-known 1/3 (stay) versus 2/3 (switch).
- More doors with only one revealed makes switching even more favorable.
- The host must always reveal a non-prize door on purpose for this math to hold.
This tool is for general educational and probability-learning purposes.