Monty Hall Switching

Lesson notes

Overview

Switching doors wins two thirds of the time — a counter-intuitive Bayesian result. This topic is rated advanced and is number 243 of 245 in Mathematics. Deeper models, formal results and harder reasoning.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

In the Monty Hall problem you pick one of three doors, one hiding a car, and the host, who knows where it is, opens a different door showing a goat and offers you a switch. Switching wins with probability 2/3, because your first pick had only a 1/3 chance and the host’s action concentrates the remaining 2/3 on the other closed door. The result startles many people, including mathematicians, and is well tested by simulation.

Reading the relationship

The expression "P(win | switch) = 2/3" is a compact summary of the idea. It uses the symbols P, w, i, n, s, t; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Gambler’s Ruin”, the topic before it in Mathematics. Once you are comfortable, continue with “Regression to the Mean”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.