HMMT 二月 2002 · ADV 赛 · 第 10 题

HMMT February 2002 — ADV Round — Problem 10

个人笔记

One fair die is rolled; let a denote the number that comes up. We then roll a dice; letthe sum of the resulting a numbers be b . Finally, we roll b dice, and let c be the sum of theresulting 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 ofthe 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 sumis (7 / 2) b . Likewise, when we roll a dice, the expected value of their sum b is (7 / 2) a , so theexpected value of c is (7 / 2) a . Similar reasoning again shows us that the expected value of 2
a is 7 / 2 and so the expected value of c overall is (7 / 2) = 343 / 8.3 3