HMMT 二月 2005 · GEN1 赛 · 第 6 题
HMMT February 2005 — GEN1 Round — Problem 6
题目详情
英文原题
- In an election, there are two candidates, A and B , who each have 5 supporters. Eachsupporter, independent of other supporters, has a probability of voting for his or her 1
candidate and a probability of being lazy and not voting. What is the probability of 12
a tie (which includes the case in which no one votes)?2
⌊ ⌋ ⌊ ⌋ ⌊ ⌋
a + b b + c c + a
解析
英文解析
- In an election, there are two candidates, A and B , who each have 5 supporters. Eachsupporter, independent of other supporters, has a probability of voting for his or her 1
candidate and a probability of being lazy and not voting. What is the probability of 12
a tie (which includes the case in which no one votes)?2
Solution: 63 / 256
The probability that exactly k supporters of A vote and exactly k supporters of B vote
( )
is · . Summing over k from 0 to 5 gives 152
k 210
( )
1 252 63
(1 + 25 + 100 + 100 + 25 + 1) = = .
2 1024 25610
⌊ ⌋ ⌊ ⌋ ⌊ ⌋
a + b b + c c + a