The Monty Hall Problem: Why You Should Always Switch Doors

The famous probability puzzle that baffled PhDs and why switching doubles your chances.

The Game Show Premise

You are on a game show with 3 closed doors. Behind one door is a sports car; behind the other two are goats. You choose Door 1. The host (who knows what is behind each door) opens Door 3 to reveal a goat, and asks: 'Do you want to switch to Door 2?'

Initial Choice: 1 in 3 chance (33.3%) of picking the car

Why Common Intuition Is Wrong

Most people assume a 50/50 probability because two doors remain. However, the host's action is constrained: they must ALWAYS open a door with a goat.

Key Insight: The host provides new information about the unchosen door.

The Mathematical Proof

• P(Car behind Door 1 initially) = 1/3 • P(Car behind Door 2 or 3 combined) = 2/3 • Since Door 3 is revealed to be empty, the entire 2/3 probability concentrates onto Door 2.

Winning Probability: Staying = 33.3% | Switching = 66.7% (2x advantage)

Simulation Verification

Computer simulations across 1,000,000 iterations consistently confirm that players who switch win the car 666,666 times.

Empirical Result: P(Win with Switch) = 0.6667

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