HMMT 二月 2003 · COMB 赛 · 第 1 题

英文解析

1. You have 2003 switches, numbered from 1 to 2003, arranged in a circle. Initially, eachswitch is either ON or OFF, and all configurations of switches are equally likely. Youperform the following operation: for each switch S , if the two switches next to S wereinitially in the same position, then you set S to ON; otherwise, you set S to OFF.
   What is the probability that all switches will now be ON?
   2002
   Solution: 1 / 2
   2003
   There are 2 equally likely starting configurations. All switches end up ON if andonly if switches 1 , 3 , 5 , 7 , . . . , 2003, 2 , 4 , . . . , 2002 — i.e. all 2003 of them — were initiallyin the same position. This initial position can be ON or OFF, so this situation occurs
   2003 2002
   with probability 2 / 2 = 1 / 2 .