The dice sum to 8, is a 5 hiding in there?

Question

Behind a screen, a friend rolls two fair dice and announces only that the two faces sum to 8. You cannot see the dice. Given that the sum is 8, what is the probability that at least one die shows a 5?

Accepted Answer

Condition on the information you have: the sum is 8. List the ordered outcomes that sum to 8: (2,6),\ (3,5),\ (4,4),\ (5,3),\ (6,2). There are 5 equally likely outcomes in this restricted world. Which of them include a 5? (3,5) 	ext{ and } (5,3), exactly 2 of them. Therefore P(	ext{a 5} \mid 	ext{sum} = 8) = \frac{2}{5} = 0.4. Takeaway. Conditioning shrinks the sample space to only the outcomes consistent with the given fact, and within that smaller space the outcomes remain equally likely. Count favorable cases over the conditioned total, never over the original 36.

The dice sum to 8, is a 5 hiding in there?

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