HMMT 二月 2005 · 冲刺赛 · 第 25 题

HMMT February 2005 — Guts Round — Problem 25

[9] An ant starts at one vertex of a tetrahedron. Each minute it walks along a random edge to an adjacent vertex. What is the probability that after one hour the ant winds up at the same vertex it started at?

解析

An ant starts at one vertex of a tetrahedron. Each minute it walks along a random edge to an adjacent vertex. What is the probability that after one hour the ant winds up at the same vertex it started at? 59 59 Solution: (3 + 1) / (4 · 3 ) Let p be the probability that the ant is at the original vertex after n minutes; then n p = 1. The chance that the ant is at each of the other three vertices after n minutes is 0 1 (1 − p ). Since the ant can only walk to the original vertex from one of the three others, n 3 1 1 and at each there is a probability of doing so, we have that p = (1 − p ). Let n +1 n 3 3 1 1 1 3 q = p − . Substituting this into the recurrence, we find that q = + ( − q − ) = n n n +1 n 4 4 3 4 ( ) n 1 3 3 1 − q . Since q = , q = · − . In particular, this implies that n 0 n 3 4 4 3 59 1 1 3 1 3 + 1 p = + q = + · = . 60 60 60 59 4 4 4 3 4 · 3