PUMaC 2013 · 组合（B 组） · 第 8 题

PUMaC 2013 — Combinatorics (Division B) — Problem 8

[ 8 ] You roll three fair six-sided dice. Given that the highest number you rolled is a 5, the a expected value of the sum of the three dice can be written as in simplest form. Find a + b . b 1

解析

[ 8 ] You roll three fair six-sided dice. Given that the highest number you rolled is a 5, the a expected value of the sum of the three dice can be written as in simplest form. Find a + b . b Solution Note that this is not the same as fixing one die at 5, then randomizing the rest from 1 to 5. To compute the expected value correctly, we consider three cases: rolls that include exactly one, two, three 5s. To make counting easier we think of the rolls as happening sequentially and count each distinct sequence as an instance. Non-5 rolls are equally likely to be 1, 2, 3, 4 and are therefore worth 2.5 on average. So we have: 0 Three 5s: (1 ∗ 4 ) ∗ 15 = 15 1 Two 5s: (3 ∗ 4 ) ∗ 12 . 5 = 150 2 One 5: (3 ∗ 4 ) ∗ 10 . 0 = 480 645 We end up with 1 + 12 + 48 = 61 instances and a total sum of 645. The fraction , already 61 in simplest form, gives sum 706, and that is the answer. 2