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Monday, September 28, 2009

Book Review: The Monty Hall Problem

Humans suck at probabilities. We've mentioned that before on this blog, nothing really new there. The astounding part is the way we will go to such ends to defend our choices once we've made them. Mention this problem at any gathering or forum and I guarantee a huge discussion will ensue. It is one of the sure things.

You are a contestant at a game show. You are given a choice between three doors. One of the doors conceal a car, the other two goats. You choose a door at random. The host opens one of the other doors, revealing a goat. Then you are given the choice to switch your choice to the other still-closed door. In order to maximize your chance at winning, should you switch?

Nobody seems to be able to agree on this. (Well unless you've read this book, in which case you know the answer.) The thing is, we enter here an area where mathematical probabilities do not work the way you are used to them working, the way your brain wants them to work.

It's counterintuitive and confusing. Nothing in this problem is as common sense would try and tell you. Don't listen to your intuition. It's wrong. Only the pure math is right.

Rosenhouse goes well into detail on why we think the way we think. Then it's backed up with math - there is a fair bit of math in this book. Fortunately, most of it is high school calculus level, so it is easy to comprehend. But the real interesting part is when it talks about why our brains get such a cognitive error in calculating the probabilities for winning this game show. Come to mention it, our brains get majorly confused when trying to calculate probabilities for most events more complex than a coin flip. (And as will also be shown, even the probability of that depends on how the question is phrased)

The Monty Hall problem (the problem, not the book) has hundreds of real-world scenarios it applies to. Really, it's like the FedEx arrow - once you've seen it you will see it everywhere. In our never-ending quest for rationality, it would be good of you to learn how to spot them, and how to properly calculate them, so you can avoid falling in this very common pitfall.

Spoiler alert: You should switch doors to maximize your chance at winning.

Jason Rosenhouse also writes a blog,
EvolutionBlog, which we recommend.

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