HMMT 十一月 2014 · 冲刺赛 · 第 15 题

HMMT November 2014 — Guts Round — Problem 15

[ 9 ] Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex 1 (each with probability ) and walks along the edge to it. What is the probability that after 10 minutes, 2 he ends up where he had started? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND Organization Team Team ID# 1 1

解析

[ 9 ] Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex 1 (each with probability ) and walks along the edge to it. What is the probability that after 10 minutes, 2 he ends up where he had started? 127 Answer: Let A denote a clockwise move and B denote a counterclockwise move. We want to 512 have some combination of 10 A’s and B’s, with the number of A’s and the number of B’s differing by ( ) ( ) ( ) 10 10 10 254 127 a multiple of 5. We have + + = 254. Hence the answer is = . 10 0 5 10 2 512 1 1