I reserve the right to be wrong, given that some really smart guys see it the other way, but it seems pretty simple. There’s no difference between the remaining 2 closed doors.
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It is not independent event, because the doors still closed are not random. One of them can not be opened for a reason.
In the 3 door example, the host has no choice when picking a door to show you the goat, when your first guess is wrong.
When your first guess is wrong (2/3 chance), the hose must opened (100% chance) a certain door that has goat behind it, then the remaining door must have the prize (100% chance).
So, 66% * 100% * 100% = 66%
