If the host opens a door at random and happens to reveal a goat, does switching still help?

No, in that variation it doesn't matter. The reason switching helps in the standard problem is that the host's choice is constrained by his knowledge of the car's location.

What if there are 4 doors and the host opens 2 of them?

Switching still wins. With 4 doors, your initial pick has probability 1/4. The remaining unopened door has probability 3/4. The general formula: with $n$ doors and the host opening $n-2$ of them, switching wins with probability $(n-1)/n$.

Did Marilyn vos Savant respond to her critics?

Yes, in three follow-up columns. She explained the answer five different ways, ran simulations in newspapers, and eventually had elementary-school classes do the experiment with paper cups.

Was Monty Hall actually like this on the real show?

No. The real Let's Make a Deal didn't follow strict rules. The mathematical "Monty Hall problem" is a clean idealization; it doesn't describe the actual show.

Why does this still feel wrong even after the proof?

Because human intuition for probability evolved in a world without game-show hosts. We are quite good at reasoning about chance when outcomes are physical and symmetric, and quite bad at reasoning when outcomes depend on the actions of an informed agent. Run the simulator above for 1,000 rounds. Watching the numbers converge is the only known cure.

Probability

The Monty Hall Problem, Settled on This Page

Marilyn vos Savant got 10,000 letters telling her she was wrong. Nearly 1,000 of them were from people with PhDs. She wasn't wrong. Here is the proof, and a live simulator you can run a thousand times in ten seconds.

Quick answer

Always switch. Switching wins 2/3 of the time. Staying wins 1/3 of the time. The intuitive answer, that your odds are 50/50 once one door is opened, is wrong. The host's reveal carries information about where the car isn't, and that information is valuable. Switching captures it; staying doesn't. The simulator below proves it in front of your eyes in about ten seconds.

Open this in math.im for the full interactive version →

Canonical URL: https://math.im/blog/monty-hall-problem.html

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