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集齐 1..5 优惠券：期望盒数

Collecting Coupons

专题
Probability / 概率
难度
L4

题目详情

Coupons in cereal boxes are numbered 1 to 5, and a set of one of is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set?

解析

这是 coupon collector。

集齐 pp 种的期望为 pHppH_p，其中 Hp=∑k=1p1kH_p=\sum_{k=1}^p\frac{1}{k}。

取 p=5p=5：

E[N]=5(1+12+13+14+15)=5⋅13760=13712≈11.42.\mathbb{E}[N]=5\left(1+\frac12+\frac13+\frac14+\frac15\right)=5\cdot\frac{137}{60}=\boxed{\frac{137}{12}}\approx 11.42.