收集优惠券

Collecting Coupons

专题: Probability / 概率
难度: L2
来源: OpenQuant

题目详情

麦片盒中的优惠券编号为 1 到 5，集齐每种各一张就能兑奖。每盒只有一张优惠券，平均需要买多少盒才能集齐一套？

Coupons in cereal boxes are numbered 1 to 5, and a set of one of each 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

第一盒一定会得到一个编号。接下来，再抽到一个新编号的概率依次是 $\frac{4}{5}, \frac{3}{5}, \frac{2}{5}, \frac{1}{5}$ ，因此拿到第 2、3、4、5 个新编号所需盒数的期望分别为 $\frac{5}{4}, \frac{5}{3}, \frac{5}{2}, \frac{5}{1}$ 。所以集齐一套所需盒数的平均值为：

5(\frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{2} + 1) \approx 11.42

Original Explanation

We get one of the numbers in the first box. Now the chance of getting a new number from the next box is $\frac{4}{5}$ . The second new number requires $1 / (4/5) = \frac{5}{4}$ boxes. The third number requires an additional $1 / (3/5) = \frac{5}{3}$ ; the fourth $\frac{5}{2}$ , the fifth $\frac{5}{1}$ . Thus the average number of boxes required is:

5(\frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{2} + 1) \approx 11.42