The Monty Hall Problem

Probability · Axiom Academy

EXAMPLE The Monty Hall Problem One of probability's most famous paradoxes Setup: You're on a game show with three doors. Behind one door is a car, behind the other two are goats. Step 2: The host (who knows what's behind each door) opens Door 3, revealing a goat. Step 3: The host offers you a choice: stick with Door 1, or switch to Door 2. Question: Should you switch? Does it matter? Excellent work! You've completed this example. Here's what we learned: Counterintuitive result: Switching doubles your chances of winning! You win 2/3 of the time by switching, versus only 1/3 by staying. Why it works: When you initially pick, you have a 1/3 chance of being right and 2/3 chance of being wrong. The host's action doesn't change YOUR door's probability, but it eliminates one wrong option from the other two doors. The key insight: If you picked wrong initially (2/3 chance), the host eliminates the other wrong door, so switching guarantees you win. If you picked right initially (1/3 chance), switching makes you lose. Information matters: The host provides information by deliberately revealing a goat. This isn't a random event—the host KNOWS and chooses strategically. All three scenarios: Car behind Door 1 (you picked right, switching loses); Car behind Door 2 (you picked wrong, switching wins); Car behind Door 3 (you picked wrong, switching wins). That's 2 out of 3 scenarios where switching wins!

This is the written version of the interactive lesson above. See the full Probability course.