HMMT 二月 2010 · 冲刺赛 · 第 31 题

英文解析

31. [ 21 ] In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. Ifeach person votes for a single candidate at random, what is the expected number of candidates thatwill 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
    that a chosen candidate will receive at least one vote is 1 − ( ) = . By linearity of expectations,5
    5 3125
    2101 2101
    the final answer is this probability times the number of candidates, or 5 · = .
    3125 625