the intuition everyone has wrong: "after a door is eliminated, it's 50-50 between the two remaining." this feels obviously right. it is not.
what actually happens: when you first pick, the probability the car is behind your door is 1/3. the probability the car is behind the other two doors collectively is 2/3.
when the host opens one of the two other doors (and crucially, reveals a goat — the host knows), that 2/3 probability collapses onto the single remaining unopened door. you didn't learn anything about your original choice. you learned everything about the other two.
switching wins 2/3 of the time. staying wins 1/3.
if you're still skeptical, imagine 100 doors. you pick one. the host opens 98 goat doors. now switch. feels obvious, right? it's the same game.
— in 1990, marilyn vos savant published this solution in parade magazine. she received ~10,000 angry letters. ~1,000 were from PhDs. all wrong.