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生日放假：最大化期望工日

Birthday Holidays 2

专题
Probability / 概率
难度
L4

题目详情

The following problem is described in Cacoullos (1989, pp. 35- 36).

A worker's legal code specifies as a holiday any day during which at least one worker in a certain factory has a birthday. All other days are working days. How many workers (n) must the factory employ so that the expected number of working man- days is maximized during the year?

解析

设 nn 名工人生日独立均匀。某天为工作日概率为 (364/365)n(364/365)^n，所以期望工日数

F(n)=n⋅365(364365)n.F(n)=n\cdot 365\left(\frac{364}{365}\right)^n.

连续近似下最优满足 1n+ln⁡(364/365)=0\frac{1}{n}+\ln(364/365)=0，因此

n≈−1ln⁡(364/365)≈365,\boxed{n\approx -\frac{1}{\ln(364/365)}\approx 365},

最优整数在 364 或 365 附近。