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## Main Question or Discussion Point

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 one's sight and some honest person who can see the result tells us that at least one of the dice came up 6. This restricts the total number of possible combinations to the following 11 pairs:

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

of which exactly two sum to 7. So we would now calculate the probability of a 7 as 2/11, slightly more than 1/6. Of course, the same analysis would give a probability of 2/11 if our honest friend had sais 'at least one 5' instead of 'at least one 6'. Or if he had said 'at least one 4'. In fact, we would arrive at the same probability if he had said '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 at fist without waiting for our friend to tell us? It would seem that this improves our a priori odds of rolling a 7 from 1/6 to 2/11.

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

of which exactly two sum to 7. So we would now calculate the probability of a 7 as 2/11, slightly more than 1/6. Of course, the same analysis would give a probability of 2/11 if our honest friend had sais 'at least one 5' instead of 'at least one 6'. Or if he had said 'at least one 4'. In fact, we would arrive at the same probability if he had said '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 at fist without waiting for our friend to tell us? It would seem that this improves our a priori odds of rolling a 7 from 1/6 to 2/11.