Being a backgammon player, I prefer the puzzle in the following format:

The probability of rolling a 7 with a pair of ordinary dice is 1/6 because, of the 36 possible (and equally likely) combinations, six of them sum to 7. Now, suppose the dice are rolled out of sight, and some honest person who can see the results tells us that at least one of the dice came up 6. This restricts the total number of possible combinations to the following eleven pairs:

(1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (6,1) (6,2) (6,3) (6,4) (6,5)

of which exactly two sum to 7. So we would now assess the probability of a 7 as 2/11, just slightly better than 1/6. Of course, the same analysis would give a probability of 2/11 if our honest friend had reported "at least one 5" instead of "at least one 6". Or if he had reported "at least one 4". In fact, we would arrive at the same probability if he reported "at least one n" for ANY value of n. Obviously we have "at least one n" for SOME value of n, so why not just assume this without waiting for our friend to tell us? It would seem that this magically improves our a priori odds of rolling a seven from 1/6 to 2/11.

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