Question

一所学校有 $100$ 名学生和 $5$ 名老师。第一节课中，每名学生上一门课，每名老师教一门课。五门课的学生人数分别为 $50, 20, 20, 5$ 和 $5$ 。若随机选一名老师并记录其班级人数，所得平均值为 $t$ 。若随机选一名学生并记录其所在班级人数，包括该学生本人，所得平均值为 $s$ 。求 $t-s$ ？

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5.$ Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s?$

$-18.5$

$-13.5$

$0$

$13.5$

$18.5$

答案：B

知识点：期望值平均数

难度评级：1140

解答：

期望值公式为 $\mathbb E[X]=\sum_x x\,\Pr(X=x).$

因此且 $t = \dfrac{1}{5} (50 + 20 + 20 + 5 + 5)$ $= \dfrac{1}{5} \cdot 100 = 20$ $s = 50 \cdot \dfrac{50}{100} + 20 \cdot \dfrac{20}{100}$ $+ 20 \cdot \dfrac{20}{100} + 5 \cdot \dfrac{5}{100} + 5 \cdot \dfrac{5}{100}$ $= 25 + 4 + 4 + .25 + .25 = 33.5$

$t - s = -13.5$ 。

所以正确答案是 B。

Recall the expected-value formula $\mathbb E[X]=\sum_x x\,\Pr(X=x).$

Therefore, $t = \dfrac{1}{5} (50 + 20 + 20 + 5 + 5)$ $= \dfrac{1}{5} \cdot 100 = 20$ and $s = 50 \cdot \dfrac{50}{100} + 20 \cdot \dfrac{20}{100}$ $+ 20 \cdot \dfrac{20}{100} + 5 \cdot \dfrac{5}{100} + 5 \cdot \dfrac{5}{100}$ $= 25 + 4 + 4 + .25 + .25 = 33.5$

$t - s = -13.5.$

Thus, B is the correct answer.

2021 AMC 10A Fall 第 10 题

答案：B

难度评级：1140

解答：

其他年份的第 10 题