HMMT 二月 2014 · 冲刺赛 · 第 2 题

HMMT February 2014 — Guts Round — Problem 2

[ 4 ] Let x , . . . , x be defined so that for each i , x is a (uniformly) random integer between 1 and 6 1 100 i inclusive. 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.

解析

[ 4 ] Let x , . . . , x be defined so that for each i , x is a (uniformly) random integer between 1 and 6 1 100 i inclusive. 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, i 1 50 the expected value is 100 · = . 6 3