Probability two dice sum to seven

Question

You roll two fair six-sided dice. What is the probability that the faces sum to 7? Is 7 the most likely total?

Accepted Answer

Treat the dice as distinguishable (say, one red and one blue). Then there are 6 	imes 6 = 36 equally likely ordered outcomes. The outcomes summing to 7 are (1,6),\ (2,5),\ (3,4),\ (4,3),\ (5,2),\ (6,1), which is 6 outcomes. So P(	ext{sum} = 7) = \frac{6}{36} = \frac{1}{6}. Is 7 the most likely total? Yes. The number of ordered outcomes giving a sum s is 6 - s - 7 for s = 2, \dots, 12: one way to make 2 or 12, two ways to make 3 or 11, and so on, peaking at six ways to make 7. Every other total has strictly fewer combinations, so 7 is the unique mode. Takeaway. Always count ordered outcomes …

Probability two dice sum to seven

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