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五封求职信全装错信封的概率

Application Letters

专题
Probability / 概率
难度
L4

题目详情

你有 5 封互不相同的求职信要寄给 5 家不同公司，对应有 5 个信封（分别写着不同公司的地址）。

一个 3 岁小孩把每封信随机塞进一个信封（完全不考虑匹配）。

问：5 封信全部装进“错误信封”的概率是多少？

You have 5 distinct job-application letters to 5 distinct firms. You have 5 envelopes, each addressed to a different firm. A 3-year-old child randomly put one letter into each envelope, disregarding any matching. What is the probability that all 5 letters end up in the wrong envelopes?

解析

这是 5 个元素的错排（derangement）概率。

错排数为 !5=44!5=44，总排列数为 5!=1205!=120，因此概率为

!55!=44120=1130.\frac{!5}{5!}=\frac{44}{120}=\frac{11}{30}.

Original Explanation

This is the “derangement” problem for 5 items. Let EiE_i be the event “the ii-th letter is in the correct envelope.” By Inclusion-Exclusion, the probability that none is correct is 1  −  (1−12!+13!−14!+15!)  =  1130.1 \;-\; \Bigl(1 - \frac{1}{2!} + \frac{1}{3!} - \frac{1}{4!} + \frac{1}{5!}\Bigr) \;=\; \frac{11}{30}.