The two dice differ, what is the sum?

Question

You roll two fair dice. You are told, truthfully, that the two dice show different numbers. Given that, what is the probability that they sum to 5?

Accepted Answer

Start from all 36 ordered outcomes, then keep only those consistent with the clue "different numbers." The six doubles (1,1),(2,2),\dots,(6,6) are excluded, leaving 36 - 6 = 30 equally likely outcomes. Among those 30, which sum to 5? A sum of 5 comes from (1,4),(4,1),(2,3),(3,2), and all four of these already have two different faces, so all four survive the conditioning. Therefore P(	ext{sum} = 5 \mid 	ext{different faces}) = \frac{4}{30} = \frac{2}{15} \approx 0.133. The clue helped a little: unconditionally the chance of a sum of 5 is 	frac{4}{36} = 	frac19, and since a sum of 5 can ne…

The two dice differ, what is the sum?

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