生日放假：最大化期望工日

Birthday Holidays

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Labor laws in Erewhon require factory owners to give every worker a holiday whenever one of them has a birthday and to hire without discrimination on grounds of birthdays. Except for these holidays they work a 365- day year. The owners want to maximize the expected total number of mandays worked per year in a factory. How many workers do factories have in Erewhon?

解析

设工厂雇佣 $n$ 人，生日独立均匀分布在 365 天。

某天为“工作日”当且仅当没人生日在这天，其概率为

\left(\frac{364}{365}\right)^n.

因此期望工作天数为 $365\left(\frac{364}{365}\right)^n$ ，期望“工日数”（man-days）为

F(n)=n\cdot 365\left(\frac{364}{365}\right)^n.

忽略整数约束，把 $a=\frac{364}{365}$ ，最大化 $n a^n$ ：

\frac{d}{dn}\ln(n a^n)=\frac{1}{n}+\ln a=0 \Rightarrow n^*= -\frac{1}{\ln a}.

由于 $\ln\left(\frac{364}{365}\right)\approx -\frac{1}{365}$ ，得到

\boxed{n\approx 365}.

取整数时最优在 364 或 365 附近。