HMMT 二月 2002 · ADV 赛 · 第 10 题

HMMT February 2002 — ADV Round — Problem 10

One fair die is rolled; let a denote the number that comes up. We then roll a dice; let the sum of the resulting a numbers be b . Finally, we roll b dice, and let c be the sum of the resulting b numbers. Find the expected (average) value of c . 1

解析

One fair die is rolled; let a denote the number that comes up. We then roll a dice; let the sum of the resulting a numbers be b . Finally, we roll b dice, and let c be the sum of the resulting b numbers. Find the expected (average) value of c . Solution: 343 / 8 The expected result of an individual die roll is (1+2+3+4+5+6) / 6 = 7 / 2. For any particular value of b , if b dice are rolled independently, then the expected sum is (7 / 2) b . Likewise, when we roll a dice, the expected value of their sum b is (7 / 2) a , so the 2 expected value of c is (7 / 2) a . Similar reasoning again shows us that the expected value of 3 a is 7 / 2 and so the expected value of c overall is (7 / 2) = 343 / 8. 3