集齐 5 种玩具

Collecting Toys I

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

每盒麦片里随机附带 5 种不同玩具中的 1 种，且各玩具等概率、独立。问：平均需要开多少盒才能集齐全部 5 种玩具？

Every box of cereal contains one toy from a group of five distinct toys, each of which is mutually independent from the others and is equally likely to be within a given box. On average, how many boxes of cereal will you need to open in order to collect all five toys?

解析

这是优惠券收集问题：期望为

5\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)=\frac{137}{12} \approx 11.4167.

Original Explanation

Choosing a distinct toy on each new box follows a geometric distribution. For the first distinct toy, we know that the first box we open will be a new toy. For the second distinct toy, there is a $\frac{4}{5}$ probability of receiving a new toy; thus, it should take $\frac{5}{4}$ boxes on average to receive a new toy. This logic follows for the third ( $\frac{5}{3}$ ), fourth ( $\frac{5}{2}$ ), and fifth ( $\frac{5}{1}$ ) toy. Hence, the total number of boxes we opened on average to collect all five toys is:

1 + \frac{5}{4} + \frac{5}{3} + \frac{5}{2} + \frac{5}{1} = \frac{137}{12}