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?'

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.

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.

Simulation Verification

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

4. The Monty Hall Problem Explained

In a game show with 3 doors (1 car, 2 goats), you pick Door 1 (1/3 chance of car). Host Monty opens Door 3 showing a goat. Should you switch to Door 2?

5. Bayes' Theorem & Conditional Probability

Bayes' Theorem updates probability estimates as new evidence arrives: P(A|B) = P(B|A)P(A) / P(B).

6. The Birthday Paradox: Probability of Shared Birthdays

In a group of just 23 randomly chosen people, the probability that at least two people share the exact same birthday exceeds 50%! In a group of 57 people, the probability jumps to 99%.

7. The Gambler's Fallacy & Law of Large Numbers

The Gambler's Fallacy is the mistaken belief that past independent random events affect future outcomes (e.g. thinking a coin that landed heads 5 times must land tails next). The Law of Large Numbers dictates that empirical averages converge to expected values only over long runs.

8. The Gambler's Ruin & Random Walk Math

A gambler starting with $N$ dollars betting $1 on fair coin flips will eventually hit bankruptcy ($0) unless their opponent has finite capital.

9. Expected Value & Insurance Risk Pricing

Insurance companies set premiums by calculating Expected Value $E(X) = \sum x_i P(x_i)$ across large policy populations.