HMMT 十一月 2014 · 冲刺赛 · 第 15 题
HMMT November 2014 — Guts Round — Problem 15
题目详情
- [ 9 ] Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex
(each with probability ) and walks along the edge to it. What is the probability that after 10 minutes,1
he ends up where he had started?2
HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND
Organization Team Team ID#
1 1
英文原题
[ 9 ] Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex
(each with probability 1
2 ) and walks along the edge to it. What is the probability that after 10 minutes,
he ends up where he had started?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 9 ] Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex
(each with probability ) and walks along the edge to it. What is the probability that after 10 minutes,1
he ends up where he had started?2
Answer: Let A denote a clockwise move and B denote a counterclockwise move. We want to 127
have some combination of 10 A’s and B’s, with the number of A’s and the number of B’s differing by 512
( ) ( ) ( )
10 10 10
254 127
a multiple of 5. We have + + = 254. Hence the answer is = .
0 5 10 2 51210
1 1