In the Monty Hall Problem, there are three doors. Let's take two scenarios. In both the host reveals that there is a goat behind door number two. In the first scenario, I choose the first door. According to the "correct" solution, after the host opens door 2, door 1 has a 1/3 chance of having the car, and door 3 2/3. But say I choose door 3. Then door 3 has a 1/3 chance, and door 1 2/3. How can I get two different answers, differing solely based on my initial choice?
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The two scenarios you describe both exclude the possibility, which was present at the beginning, that the car is behind the middle door. This cuts out one third of the sample space and leaves the two other possibilities equal.
If the car had been behind the middle door, one of the other doors would have been opened, and you would have needed to swap to win it.