HMMT 二月 2017 · COMB 赛 · 第 1 题

HMMT February 2017 — COMB Round — Problem 1

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled 0 , 1 , 2 , . . . , 9 . First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

解析

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled 0 , 1 , 2 , . . . , 9 . First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll? Proposed by: Yang Liu 11 Answer: 200 3 First, there are 10 = 1000 triples ( a, b, c ) . Now, we should count how many of these triples satisfy a + b = c. If c = 0, we get 1 triple (0 , 0 , 0). If c = 1, we get two triples (1 , 0 , 1) and (0 , 1 , 1) . Continuing, this gives that the total number of triples is 1 + 2 + · · · + 10 = 55 . Therefore, our final answer is 55 11 = . 1000 200