# Monty Hall Problem

**Type:** Puzzle  
**Domain:** Named Results / Probability  
**Codex URL:** /mathematics/named-results/monty-hall-problem/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Switching doors doubles winning odds (2/3 vs 1/3) — famously counterintuitive.

## The setup
3 doors, 1 car. Pick one; host (who knows where the car is) opens an empty door among the remaining two; offered the chance to switch.

## Why switch: case-by-case
Car behind your pick (1/3): switching loses. Car behind either other door (2/3 total): switching wins. Switching wins 2/3 of the time.

## Critical assumption
Host must always know the car's location and always open an empty door — changing this behavior changes the odds.

## 1990 controversy
Marilyn vos Savant's Parade column sparked ~10,000 letters, many from PhDs insisting the correct answer was wrong.

## Historical priority
Steve Selvin (1975, American Statistician) posed and solved an equivalent problem 15 years before vos Savant's famous column. Bertrand's box paradox (1889) is an even earlier related puzzle.

## Bayes' Theorem formalization
P(car behind original door | host opens door 3) = 1/3; P(other door | same evidence) = 2/3 — rigorous confirmation via Bayesian updating.

## Pigeon study
Herbranson & Schroeder (2010): pigeons converged on optimal switching strategy via reinforcement learning, while humans often persisted with suboptimal "stay" strategy despite explanation.

## Sources
### Tier 1
- Selvin, S. (1975). The American Statistician.
- vos Savant, M. (1990). Parade Magazine.
### Tier 2
- Herbranson, W.T. and Schroeder, J. (2010). J. Comparative Psychology.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*