HMMT 二月 2005 · GEN1 赛 · 第 6 题

HMMT February 2005 — GEN1 Round — Problem 6

In an election, there are two candidates, A and B , who each have 5 supporters. Each 1 supporter, independent of other supporters, has a probability of voting for his or her 2 1 candidate and a probability of being lazy and not voting. What is the probability of 2 a tie (which includes the case in which no one votes)? ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ a + b b + c c + a

解析

In an election, there are two candidates, A and B , who each have 5 supporters. Each 1 supporter, independent of other supporters, has a probability of voting for his or her 2 1 candidate and a probability of being lazy and not voting. What is the probability of 2 a tie (which includes the case in which no one votes)? Solution: 63 / 256 The probability that exactly k supporters of A vote and exactly k supporters of B vote ( ) 2 5 1 is · . Summing over k from 0 to 5 gives 10 k 2 ( ) 1 252 63 (1 + 25 + 100 + 100 + 25 + 1) = = . 10 2 1024 256 ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ a + b b + c c + a