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The Monty Hall Problem

Should you switch doors? The answer surprises everyone.

Welcome to Let's Make a Deal!
Pick a door to begin.
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2
3

๐Ÿ”„ When You Switched

Games Played: 0
Wins: 0
Win Rate: โ€”

๐Ÿšช When You Stayed

Games Played: 0
Wins: 0
Win Rate: โ€”

๐Ÿ”ฌ Monte Carlo Simulation

Run thousands of games to see the true probabilities emerge!

Always Switch Strategy

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Run simulation to see results

Always Stay Strategy

โ€”
Run simulation to see results

Why Switching Wins 2/3 of the Time

When Marilyn vos Savant published the correct solution in 1990, she received over 10,000 letters telling her she was wrongโ€”including nearly 1,000 from PhD holders! Even the legendary mathematician Paul Erdล‘s refused to believe it until he saw a computer simulation.

The Key Insight: When you first pick a door, you have a 1/3 chance of being right. That means there's a 2/3 chance the car is behind one of the other doors. When Monty opens a door showing a goat, that 2/3 probability doesn't disappearโ€”it all concentrates on the remaining door!

The Simple Proof

Assume you always pick Door #1. Here are all possible scenarios:

๐Ÿš— Car behind Door 1

Monty opens Door 2 or 3

Switch โ†’ You lose

LOSE

๐Ÿš— Car behind Door 2

Monty must open Door 3

Switch โ†’ You WIN Door 2

WIN

๐Ÿš— Car behind Door 3

Monty must open Door 2

Switch โ†’ You WIN Door 3

WIN

Switching wins in 2 out of 3 cases = 66.7%
Staying wins in 1 out of 3 cases = 33.3%

The 1990 Controversy

Marilyn vos Savant, listed in the Guinness Book of World Records for highest IQ (228), answered this puzzle correctly in her Parade magazine column. The backlash was extraordinary:

"You blew it! Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills."
โ€” PhD mathematician, one of thousands who were wrong

Vos Savant was right. The flood of angry letters from academics became a famous case study in how even experts can be fooled by probability problems.

Why Our Intuition Fails

Our brains make two critical errors:

  1. Equiprobability Bias: We assume the two remaining doors must have equal probability (50/50). But Monty's action isn't randomโ€”he's forced to reveal a goat, which transfers probability to the door you didn't pick.
  2. Ignoring Prior Probability: We forget that our initial choice only had a 1/3 chance of being correct. That 1/3 doesn't change just because Monty opened a door.
The Bathtub Analogy: Imagine 100 doors. You pick one. Monty opens 98 doors, all showing goats, leaving your door and one other. Would you switch now? Of course! Your door still has only a 1/100 chance; the other has 99/100. The 3-door version works the same way, just with smaller numbers.