this post was submitted on 19 Aug 2023
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that's a really good question. the probability getting locked in is very counterintuitive. i think it helps to think about what happens in each case.
let's say you pick one of the 99 doors with goats behind them. this happens 99% of the time. the host is tasked with opening 98 doors. of the 99 doors you didn't choose, one has a car behind them. the host does not have a choice about which doors to open, his hand is forced. in this sense, the 99 other doors are tied together: since you originally chose a door with a goat behind it, the host is forced to leave only the door with a car unopened. so, you switch and you get a car.
next, let's say you get lucky and pick the door with a car behind it. this happens 1% of the time. now the host gets a choice about which doors to open: he gets to pick one door with a goat to leave unopened. in this scenario switching gets you no car.
so, 99% of the time, switching gets you a car. i hope this is helpful!
This definitely helped me look at it as a whole, and definitely started me down the right path of getting it. Thanks!