Question

从集合 $\{1, 2, 3, \ldots, 30\}$ 中随机选出 $5$ 个不同整数组成一个子集。这个子集中相邻整数对的平均个数是多少？（例如集合 $\{1, 17, 18, 19, 30\}$ 有 $2$ 对相邻整数。）

What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{1, 2, 3, \ldots, 30\}?$ (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)

$\dfrac{2}{3}$

$\dfrac{29}{36}$

$\dfrac{5}{6}$

$\dfrac{29}{30}$

$1$

答案：A

知识点：期望值无放回抽样

难度评级：2190

解答：

对 $29$ 个相邻对中的每一对 $(i, i+1)$ ，设指标变量在两数都被选中时为 $1$ 。这个事件的概率为 $\dfrac{5}{30} \cdot \dfrac{4}{29} = \dfrac{2}{87}$ 。

由期望的线性性，相邻整数对的期望数为 $29 \cdot \dfrac{2}{87} = \dfrac{2}{3}$ 。

所以正确答案是 A。

For each of the $29$ adjacent pairs $(i, i+1),$ let an indicator be $1$ if both are in the subset. The probability of this is $\dfrac{5}{30} \cdot \dfrac{4}{29} = \dfrac{2}{87}.$

By linearity of expectation, the expected number of consecutive pairs is $29 \cdot \dfrac{2}{87} = \dfrac{2}{3}.$

Thus, the correct answer is A.

2021 AMC 12B Fall 第 23 题

答案：A

难度评级：2190

解答：

其他年份的第 23 题