HMMT 十一月 2010 · 团队赛 · 第 4 题

英文解析

4. [ 6 ] In terms of k , for k > 0, how likely is it that after k minutes Sherry is at the vertex opposite the vertex where she started?
   1 1
   Answer: + Take p ( n ) from the last problem. By examining the last move of Sherry, thek
   6 3( − 2)
   probability that she ends up on the original vertex is equal to the probability that she ends up on the
   1 − p ( n )
   top vertex, and both are equal to for n ≥ 1.
   From the last problem,2
   p ( n )
   p ( n + 1) = 1 −
   ( )2
   2 1 2
   p ( n + 1) − = − p ( n ) −
   3 2 3
   ( )
   2 1 2 2 1 nand so p ( n ) − is a geometric series with ratio − . Since p (0) = 0, we get p ( n ) − = − − , or
   3 2 3 3 2
   ( )
   2 2 1 nthat p ( n ) = − − .
   3 3 2
   Now, for k ≥ 1, we have that the probability of ending up on the vertex opposite Sherry’s initial vertex
   ( )
   ( ) ( )
   k k
   1 − p ( k )
   1 1 2 2 1 1 1 1 1 1
   after k minutes is = − − − = + − = + .
   2 2 2 3 3 2 6 3 2 6 3( − 2)k
   Circles in Circles