this post was submitted on 19 Aug 2023
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i thought it would be helpful to make a comment on why the probability after switching is 66%, since it's very counterintuitive. the main reason has to do with why a certain door was chosen to be opened.
in more detail: it helps to instead think about a situation where there are 100 doors, 99 of which have a goat behind them. (original problem was about goats and a car). you choose one of 100 doors. there's a 1% chance you picked the door with the car. in other words, there is a 99% chance the car is behind a door you did not pick. put differently, 99 times out of 100, you are in a situation where the car is behind a door you did not choose.
afterward you pick your door, the host picks 98 doors with goats behind them and opens them. (this is another crucial detail, the host can only open doors that don't have a car behind them.) it is still true that 99% of the time, the car is behind a door you did not choose. this is because the doors were opened after you made your choice. but now, there is only one door you did not choose, so that door has a 99% chance of having a car behind it.
I never understood why in the 100-door case, the host opens 98 doors, and not just one door. That feels like changing the rules.
I fully understand the original problem with 3 doors; I know the win probability is 2/3 if you change. But whenever I hear the explanation for 100 doors case, it just made everything confusing. By opening 98 doors, it feels like the host wants you to switch to the other door. In 3 doors case it's more natural.
Because the problem is explicitly about the choice between two doors. You have to eliminate all but two choices.
But even then, you'd still have a better chance by switching.
Your intuition about the change is the whole point - it exposes why the result is what it is.
In both cases the host opens every door but one.