Probability two dice sum to eight

Question

You roll two fair six-sided dice. What is the probability the faces sum to 8? Is 8 more or less likely than 6?

Accepted Answer

Treat the dice as distinguishable (say one red, one blue), giving 6 	imes 6 = 36 equally likely ordered outcomes. The ordered outcomes summing to 8 are (2,6),\ (3,5),\ (4,4),\ (5,3),\ (6,2), which is 5 outcomes. So P(	ext{sum} = 8) = \frac{5}{36}. Eight versus six? They are equally likely. The count of ordered outcomes for sum s is 6 - s - 7, which is symmetric around 7. Both 6 and 8 sit one step from 7, so each has 6 - 1 = 5 outcomes and probability 5/36. In general a total s and its mirror 14 - s always share the same probability. The trap is treating (2,6) and (6,2) as one outcome; count…

Probability two dice sum to eight

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