---
title: "Monty Hall"
url: https://sharadbapat.com/experiments/lessons/monty-hall/
description: "Three doors, one car, a host who knows. Play it, run 1,000 games, then change one fact and watch the famous advantage disappear."
published: 2026-09-24
author: Sharad Bapat
---

[Interactive lessons](https://sharadbapat.com/experiments/lessons/)

# Monty Hall

Three doors, one car, and a host who knows where it is. Should you switch? Predict first, then play, then change one fact and watch the answer change with it. About five minutes.

## 1. Play

A car is behind one of three doors, and a goat is behind each of the other two. You pick a door. The host, who knows where the car is, opens one of the two doors you didn't pick, always showing a goat. Then he offers you a choice: stick with your door, or switch to the other closed one.

Before you play: with two doors left, does switching change your chances?

Play it a few times and switching seems to win more often, but a handful of games is mostly luck.

## 2. Play 1,000 games

A few games can't settle it. Play the same 1,000 deals both ways, with the car placed at random each time.

Sticking wins about 1 game in 3. Switching wins about 2 in 3: twice as often. Two doors are left, but it isn't 50/50.

## 3. Change one fact

Now the host doesn't know where the car is. He opens one of the other two doors at random. If he happens to reveal the car, that game doesn't count. We keep only the games where he showed a goat, so everything you can see is exactly as before: your door, an open goat door, one closed door.

In the games that count, is switching still better?

When the host guesses, sticking and switching each win about half the games that count. Nothing you can see has changed. The only difference is what the host knew.

When he knows, he is forced to avoid the car, so the door he leaves shut is shut for a reason. When he guesses, it isn't.

## 4. Why: follow your first pick

Take 30 games with a host who knows, and sort them by what your first pick turned out to be.

Your first pick is right 1 time in 3. Nothing the host does changes that, because he can always find a goat to show you. So sticking wins 1 time in 3. Switching wins every time your first pick was wrong, because the host has to open the other goat door, which leaves the car behind the closed one.

P(switching wins) = P(first pick was wrong) = 2/3

This also explains step 3. When the host guesses, the games where your first pick was a goat are exactly the ones where he's likely to open the car's door. Those games get thrown out, and they're the ones switching would have won. What's left is an even split.

## 5. A hundred doors

If it still feels like a trick, scale it up. Pick one of 100 doors. The host, who knows, opens 98 goat doors and leaves exactly one other door shut.

Your door had a 1 in 100 chance, and it still does. The host had to leave one door shut out of the other 99, and he'd never leave a goat shut if the car was behind a different door. Switching wins 99 times in 100.

## 6. Something new

Three prisoners, A, B and C. One of them has been chosen at random to be pardoned. The warden knows who. A asks him to name one of the other two who won't be pardoned. The warden, who will never name the pardoned prisoner, says "B".

What are A's chances of being pardoned now, and C's?

It's the same problem. A is your first pick, and the warden is a host who knows. His answer can't change A's chance, which stays 1 in 3. B's share moves to C, who now has a 2 in 3 chance.

The rule underneath both: when someone who knows the answer is forced to avoid it, whatever they leave alone carries information. When they're guessing, it doesn't.