HMMT 二月 2010 · 冲刺赛 · 第 31 题

HMMT February 2010 — Guts Round — Problem 31

[ 21 ] In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for?

解析

[ 21 ] In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for? 2101 4 5 Answer: The probability that a chosen candidate will receive no votes at all is ( ) , or the 625 5 probability that every person will vote for someone other than that one candidate. Then the probability 4 2101 5 that a chosen candidate will receive at least one vote is 1 − ( ) = . By linearity of expectations, 5 3125 2101 2101 the final answer is this probability times the number of candidates, or 5 · = . 3125 625