PUMaC 2016 · 团队赛 · 第 11 题

PUMaC 2016 — Team Round — Problem 11

(8) Madoka chooses 4 random numbers a, b, c, d between 0 and 1. She notices that a + b + c = 1. m If the probability that d > a, b, c can be written in simplest form as , find m + n . n

解析

The answer is 1 − E (max( a, b, c )), where E denotes the expected value. We deal with only the case where a > b, c , and split it into two further cases. 1 1 4 Case 1: a < . The probability this occurs is . The expected value of a is . 2 12 9 1 1 2 Case 2: a > . The probability this occurs is . The expected value of a is . 2 4 3 ( ) 1 4 1 2 11 7 In total, E (max( a, b, c )) = 3 + = . The probability is thus , so the answer is 12 9 4 3 18 18 7 + 18 = 25 . 2 ab − 1 ab − 1