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Don’t think it applies to this because there’s no “game show host” who knows which school he is going to choose. But I like where your head’s at.
For those playing along at home, when you pick door #1 there is a 1/3 chance that you are correct and a 2/3 chance you are wrong. If the host, knowing whether 2 or 3 has the prize, shows you that 2 does not, door #1 still has a 1/3 chance — eliminating a door which the host knew was empty doesn’t change anything with respect to #1 — but door #3 represents the other 2/3ds of the odds.