The Monty Hall Problem

If you switch, you have a two-in-three chance of winning. If you don’t, you only have a one-in-three chance of winning. Therefore, you should switch.

This puzzle is usually called the Monty Hall Problem, and the answer often gets a lot of pushback. While many people chalk this up to the answer being counterintuitive, I believe it’s partly because the puzzle is often posed in an unclear way:

“Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice?”

Even though this version tells us the host knows what’s behind the doors, it doesn’t say he’s guaranteed to open a losing door. Yes, in your game he did, but the rules don’t require him to always do so. If the host could sometimes open a winning door—or choose not to open any door—then the probabilities change.