Monty Hall Problem Interactive Simulator
Should you switch doors? Simulate and find out!
Educational Tool Only: This simulator demonstrates the famous Monty Hall probability puzzle for learning purposes. It does not predict real game show outcomes and should not be used for gambling or betting decisions.
Ready to Play Monty Hall?
Choose how many doors and games to simulate, then see how often staying versus switching wins. This simulator helps you build intuition for the famous Monty Hall problem.
Classic puzzle: You pick a door. The host opens another door revealing a goat. Should you switch or stay? Most people think it's 50/50, but switching actually wins twice as often!
Understanding the Monty Hall Problem
What is the Monty Hall Problem?
The Monty Hall problem is a famous probability puzzle named after the host of the American TV game show "Let's Make a Deal." Here's the classic setup:
- You're shown three closed doors. Behind one is a prize (like a car); behind the others are goats.
- You pick a door (say, Door 1).
- The host, who knows what's behind each door, opens another door (say, Door 3) revealing a goat.
- The host then asks: "Do you want to switch to Door 2?"
Should you switch? Most people think it doesn't matter — that it's a 50/50 chance. But mathematically, you should always switch, because switching wins 2/3 of the time!
Why Does Switching Work?
The key insight is that the host's action is not random. The host always knows where the prize is and always opens a door with a goat. This changes the probabilities:
When you first pick:
- • Probability you picked the prize: 1/3
- • Probability the prize is behind one of the other doors: 2/3
After the host opens a goat door:
- • Your original pick still has 1/3 probability
- • The remaining closed door now has the entire 2/3 probability!
- • The host's reveal "concentrates" the 2/3 into one door
Building Intuition: The 100 Doors Version
Still not convinced? Imagine the same game with 100 doors instead of 3:
- You pick 1 door out of 100. Chance of being right: 1%.
- The host opens 98 other doors, all showing goats.
- Now there's just your door and one other door left.
Would you switch? Of course! Your door is almost certainly wrong (99% chance). The host has essentially told you where the prize is by eliminating 98 wrong answers.
The same logic applies to 3 doors — just less dramatically.
The General Formula
For D doors (where the host opens D-2 goat doors):
Probability of winning by staying:
1 / D
(Your original chance of picking right)
Probability of winning by switching:
(D - 1) / D
(Everything else, concentrated into one door)
Educational Tool Only: This simulator demonstrates probability concepts for learning purposes. It should not be used for gambling, betting, or any real-world wagering decisions. Real game shows may have different rules.
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Disclaimer: This simulator is for educational and entertainment purposes only. It uses pseudorandom number generation to illustrate a classic probability puzzle. Results should not be used for gambling, betting, financial decisions, or any real-world wagering. Real game shows have different rules and outcomes.