HMMT 二月 2012 · COMB 赛 · 第 2 题
HMMT February 2012 — COMB Round — Problem 2
题目详情
英文原题
Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces
numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability
that the number on Brian’s die is larger than the sum of the numbers on George’s di ce?
解析
英文解析
- Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with facesnumbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probabilitythat the number on Brian’s die is larger than the sum of the numbers on George’s dice?
Answer: Let Brian’s roll be d and let George’s rolls be x, y, z . By pairing the situation d, x, y, z 19
with 21 − d, 7 − x, 7 − y, 7 − z , we see that the probability that Brian rolls higher is the same as the 40
probability that George rolls higher. Given any of George’s rolls x, y, z , there is exactly one number
Brian can roll which will make them tie, so the probability that they tie is . So the probability that 1
1 −120
2019
Brian wins is = .
2 40