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A comment on Making the Monty Hall problem weirder but obvious.
1/3 of the time, you guess correctly on your initial guess. If you switch, you'd be wrong.
2/3 of the time, you guess incorrectly on your initial guess. If you switch, you're right.
So when switching, the expecting outcome 1/3 of the time is 0, but the expected outcome 2/3 of the time 1. (1/3)0 + (2/3)1 = 2/3
The original problem is more complicated.
1/3 of the time, you guess correctly on your initial guess. If you switch, you'd be wrong.
2/3 of the time, you guess incorrectly on your initial guess. If you switch, you're right.
So when switching, the expecting outcome 1/3 of the time is 0, but the expected outcome 2/3 of the time 1. (1/3)0 + (2/3)1 = 2/3