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Soleron said:

They key is that, out of the doors you haven't picked, they always open a door that doesn't have the prize. So opening it doesn't change the odds of it being behind {set of all doors you didn't choose}.

When you reduce the set of doors not picked by you to one (show the rest as mot having the prize), the odds of remaining door that has not been selected by you equals the odds of all the other doors you didn't pick.