The Monty Hall problem

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One of these things is not like the other

There’s a bit of a meme going about trying to explain The Monty Hall problem. It’s an interesting statistical problem because the result appears at first to be counterintuitive.

The basic formulation comes from a game show, where the final round consisted of three boxes, and behind 1 is a car, and the other 2 are goats.

If you pick the car, you keep it.

The complication in the setup comes from the host, Monty Hall, who, after you’ve picked a box, opens another box to reveal a goat, then gives you the opportunity to stick or swap. The question is, in a statistical calculation, should you switch?

The naive answer would suggest that as there’s only 2 boxes left, there’s a 50/50 chance that you’ve chosen the car. But that formulation misses the important facts that you choose a box first, and you are always shown a goat. There’s a lot of descriptions that show how the possibilities collapse with numerous branches to show that in 2/3rds of cases, it pays to switch.

There is a simpler way of thinking about the solution. In 2/3rds of cases, you choose a goat. Since Monty had to reveal a goat, changing your mind guarantees you a car in those cases. In the other 1/3rd, you picked the car in the first case.

Sometimes statistics is unintuitive because you have the wrong model, and users can’t see the wood for the decision trees. There are good reasons for some faulty models to exist, but these should be corrected and challenged.

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