HMMT 二月 2014 · 冲刺赛 · 第 2 题

英文解析

2. [ 4 ] Let x , . . . , x be defined so that for each i , x is a (uniformly) random integer between 1 and 6
   1 100 iinclusive. Find the expected number of integers in the set { x , x + x , . . . , x + x + . . . + x } that
   1 1 2 1 2 100
   are multiples of 6.
   50 1
   Answer: Note that for any i , the probability that x + x + . . . + x is a multiple of 6 is
   1 2 i
   3 6
   because exactly 1 value out of 6 possible values of x works. Because these 100 events are independent,
   1 50 ithe expected value is 100 · = .
   6 3