Three approaches to the Monty Hall problem

Approach #1: Strategies chosen in advance

The first approach is to think of it in terms of picking a strategy in advance: either you’re a sticker or a switcher. Which strategy is the best? Well, for a sticker to win they have to pick the goat car and stick with it. One in three. For a switcher to win, they have to pick a goat and switch away when the host opens the other goat. Two in three.

Approach #2: Confidence, not probability

This is really just the same thing again, but focusing on the fact that there’s a subtle difference in the way we think about probabilities versus confidences. The concept of a probability allows for things like subconscious conflation of past versus future probabilities, whereas the question “how confident are you that you picked the car?” encourages you to enumerate the possible versions of reality you could be in.

Approach #3: Wording

Ultimately I think this is one of those riddles that would kind of dissolve if it were worded more clearly. For example:

If you pick the car, the host opens a random goat and you can switch to the other goat if you want. If you pick a goat, the host opens the other goat and you can switch to the car if you want.

This makes it clear that the host is not just opening another door at random but is actually making a decision about which door to open based on which door you chose, which might head off any “but it’s still just one in three”-style objections.