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A fair six-sided die is rolled 6 times. Let E be the expected number…
Question
A fair six-sided die is rolled 6 times. Let E be the expected number of DISTINCT faces (values from 1 to 6) that appear at least once among the 6 rolls. E is a rational number m/n in lowest terms; find m+n.
✓ Verified answer: 38807checked by our engine — not a guess
Step-by-step solution
Use indicator variables and linearity of expectation. For each face value v in {1,...,6}, let I_v = 1 if v appears at least once in the 6 rolls.
P(v never appears in 6 rolls) = (5/6)^6, so P(I_v = 1) = 1 - (5/6)^6.
The number of distinct faces = I_1 + ... + I_6, so by linearity
E = 6 * (1 - (5/6)^6).
(5/6)^6 = 5^6/6^6 = 15625/46656.
1 - 15625/46656 = 31031/46656.
E = 6 * 31031/46656 = 31031/7776 (dividing 46656 by 6).
Check lowest terms: 7776 = 2^5 * 3^5; 31031 = 7 * 11 * 13 * 31 (odd, not divisible by 3), so reduced.
m+n = 31031 + 7776 = 38807.
Final answer38807
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