HMMT 二月 2005 · 冲刺赛 · 第 25 题

英文解析

25. An ant starts at one vertex of a tetrahedron. Each minute it walks along a randomedge to an adjacent vertex. What is the probability that after one hour the ant windsup 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; thennp = 1. The chance that the ant is at each of the other three vertices after n minutes is
    (1 − p ). Since the ant can only walk to the original vertex from one of the three others,10
    3 n
    1 1
    and at each there is a probability of doing so, we have that p = (1 − p ). Letn +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
    ( )
    1 3 3 1 n − q . Since q = , q = · − . In particular, this implies thatn 0 n
    3 4 4 3
    1 1 3 1 3 + 159
    p = + q = + · = .
    60 60
    60 59
    4 4 4 3 4 · 3