HMMT 二月 2005 · COMB 赛 · 第 6 题
HMMT February 2005 — COMB Round — Problem 6
题目详情
- Three fair six-sided dice, each numbered 1 through 6, are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?
解析
- Three fair six-sided dice, each numbered 1 through 6, are rolled. What is the probability that the three numbers that come up can form the sides of a triangle? Solution: 37 / 72 Denote this probability by p , and let the three numbers that come up be x , y , and z . We will calculate 1 − p instead: 1 − p is the probability that x ≥ y + z , y ≥ z + x , or z ≥ x + y . Since these three events are mutually exclusive, 1 − p is just 3 times the probability that x ≥ y + z . This happens with probability (0 + 1 + 3 + 6 + 10 + 15) / 216 = 35 / 216, so the answer is 1 − 3 · (35 / 216) = 1 − 35 / 72 = 37 / 72.