Question

Carr 老师要求学生阅读书单上 $10$ 本书中的任意 $5$ 本。Harold 从书单中随机选 $5$ 本，Betty 也随机选五本。他们恰好有 $2$ 本书都选中的概率是多少？

Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. What is the probability that there are exactly $2$ books that they both select?

$\dfrac{1}{8}$

$\dfrac{5}{36}$

$\dfrac{14}{45}$

$\dfrac{25}{63}$

$\dfrac{1}{2}$

答案：D

知识点：组合基本概率

难度评级：1140

解答：

固定 Harold 选的 $5$ 本书。Betty 从这五本中恰好选两本有 $\binom{5}{2}$ 种方式，另外三本从 Harold 没选的 $5$ 本中选，有 $\binom{5}{3}$ 种方式。

因此 Betty 恰好选中 Harold 书单中两本、书单外三本的方式数为 $\binom{5}{2}\binom{5}{3}=100$ 。 Betty 任意选择五本书共有 $\binom{10}{5}=252$ 种方式，其中有利选择有种，所以所求概率为所以正确答案是 D。 $\frac{100}{252}=\frac{25}{63}.$

Assume that Harold has already picked his $5$ books. Of these five books, there are $\binom{5}{2}$ ways that Betty can have picked exactly two of the same books as Harold, and $\binom{5}{3}$ ways that Betty can choose her other three books from the $5$ books not on Harold's list.

Thus there are $\binom{5}{2}\binom{5}{3}=100$ favorable choices for Betty out of $\binom{10}{5}=252$ equally likely choices. The probability is $\frac{100}{252}=\frac{25}{63}.$ Thus, D is the correct answer.

2020 AMC 10B 第 11 题

答案：D

难度评级：1140

解答：

其他年份的第 11 题