Expected number of sevens

Question

You roll a pair of fair dice 12 times and record the sum each time. On average, how many of those 12 rolls sum to exactly 7?

Accepted Answer

First find the chance a single roll of two dice sums to 7. The sum is 7 for the ordered outcomes (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), which is 6 of the 36 equally likely outcomes: P(	ext{sum} = 7) = \frac{6}{36} = \frac{1}{6}. Now write the count of sevens over 12 rolls as a sum of 12 indicators, one per roll. By linearity of expectation, E[	ext{number of sevens}] = 12 	imes \frac16 = 2. So you expect two sevens on average, though of course any particular set of 12 rolls might have zero, one, three, or more. Takeaway. Expected count equals trials times per-trial probability. Getting the si…

Expected number of sevens

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