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And what if Monty *always* opens a door and offers you the chance
to switch but does so without knowing what is behind the doors?
Then, 1/3 of the time he shows you a car and you win a goat no
matter what you do. The other 2/3 of the time you win a car with
50% probability whether you do or don't switch. In other words,
you win 1/3 of the time. This is because Monty gives you no
useful information.