Monty Hall as investigative journalism

by Collin Lysford

In honor of one of the best jokes I’ve seen in a while, I’d like to offer a reframe of the Monty Hall problem that makes it less paradoxical. Especially because so many explainers out there just make it more confusing.

Monty Hall feels paradoxical because the host has information; we can show this by seeing what happens when we take that information away. Imagine you and your buddy stumbling blindly onto a naturally-occurring Monty Hall problem in the jungle. There are three doors, one with a car and two with a goat. You pick your favorite door. Then your buddy picks their favorite door, but they’re forced to pick a different one, because you’ve got your hand on yours already. Then they open their door. One third of the time, your buddy gets a car and you’re left with a weird choice: do you want to switch which door you’ve chosen, even though you’re definitely not getting a car, and you’re just picking between two goats you can’t see? ¹ But the other 2/3rds of the time, you either picked the right door or you didn’t, and switching makes no difference. You’re perfectly balanced between “Car if you stay, goat if you switch” and “Goat if you stay, car if you switch”. This is what most people’s intuition wants the normal Monty Hall problem to be like.

But the Monty Hall problem is different than my version because it’s not your buddy up there; it’s a game show host taking orders from on high. The boss doesn’t want the anti-climax of the host revealing a car and turning the game to Guess the Goat, so in cases where the host’s favorite door has a car behind it, the boss makes them change. This can never happen when your favorite door has the car (because you picked first), so it has the effect of taking Guess the Goat and turning it into “Goat if you stay, car if you switch”. And since “Car if you stay, goat if you switch” and “Goat if you stay, car if you switch” were perfectly balanced before, adding some more “Goat if you stay, car if you switch” into the mix makes switching a better deal.

The real lesson: if the boss is allowed to put their thumb on the scales, assuming the prize is randomly distributed makes you a chump who deserves every goat you get. Figure out where they aren’t letting you look, and assume that the treasure is a lot more likely to be hiding there than your first blind guess. Paradox, my ass! That’s just the first rule of investigative journalism.