Concepts · 3 min read

The Monty Hall Problem Explained (Why Switching Wins 2/3)

Published October 3, 2026Written and reviewed by Mateuss M.

The setup

One door hides a car, two hide goats. You choose a door. The host, who knows where the car is, always opens a different door with a goat and asks if you want to switch.

List every case

Say you always pick door 1 (the car position is what varies):

Each car position is equally likely, so staying wins in 1 of 3 cases and switching in 2 of 3.

Why it feels wrong

People think two doors remain, so it must be 50/50. But the host’s choice is not random: he must avoid the car. Your first pick has a 1/3 chance of being right, and the host’s action transfers the other 2/3 onto the one remaining door.

Exact enumeration gives: stay = 33.33%, switch = 66.67%. The probability calculator handles the single-event arithmetic.

Make it concrete

Imagine 100 doors: you pick one, and the host opens 98 goat doors. Would you switch to the last door? The logic is identical, just easier to feel.

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