在某个月，有 21 人参加了驾驶考试。随机抽取这 21 人中的一人。抽取的人是通过考试的男性的概率为 $\frac{1}{7}$ 。随机抽取这 21 人中的两人。她们都是未通过考试的女性的概率为 $\frac{1}{10}$ 。完成关于该月参加考试的 21 人的信息表。 (b) $\varepsilon = \{x:\ x\text{ is a student in a class}\}$ $A = \{x:\ x\text{ is a student who listens to pop music}\}$ $B = \{x:\ x\text{ is a student who listens to classical music}\}$ 已知 $n(\varepsilon)=35$ ， $n(A)=22$ ， $n(B)=14$ 且 $n(A\cap B')=12$ 。用语言描述集合 $A\cap B'$ 。 ... 求既听流行音乐又听古典音乐的学生人数。答 ... 求只听古典音乐的学生人数。答 ... 求既不听流行音乐也不听古典音乐的学生人数。答 ...

复合事件概率集合符号与表示韦恩图单一事件概率计算器使用列方程解应用题一元二次方程数据收集与整理数系与四则运算集合的并集与交集 English

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完整解法

2(a)

1. Find the number of men who passed the test from the given probability.

Let the number of men who passed the test be $m$ .

Since one person is selected from 21 people,
$\frac{m}{21}=\frac{1}{7}$
$m=21\times \frac{1}{7}=3$

So, $3$ men passed the test.

2. Find the number of women who failed the test using the probability of selecting two without replacement.

Let the number of women who failed the test be $w$ .

The probability that two selected people are both women who failed is
$\frac{w}{21}\times \frac{w-1}{20}=\frac{1}{10}$

So,
$\frac{w(w-1)}{420}=\frac{1}{10}$
$w(w-1)=42$

Since $7\times 6=42$ , $w=7$ .

So, $7$ women failed the test.

3. Use the total number of people to find the number of women who passed.

The known entries are:

Men passed $=3$ , men failed $=9$ , women failed $=7$ .

Total number of people is $21$ , so
$\text{women passed}=21-3-9-7$
$=2$

答案

\begin{array}{c|cc} & \text{Passed the test} & \text{Failed the test} \\ \hline \text{Men} & 3 & 9 \\ \text{Women} & 2 & 7 \end{array}

2(b)(i)

1. Interpret the meaning of $A\cap B'$.

$A$ is the set of students who listen to pop music.

$B'$ is the set of students who do not listen to classical music.

So $A\cap B'$ means students who are in $A$ and in $B'$ .

答案

Students who listen to pop music but do not listen to classical music.

2(b)(ii)

1. Find the number of students who listen to both pop and classical music.

Since $n(A)=22$ and $n(A\cap B')=12$ ,
$n(A\cap B)=22-12=10.$

答案

$10$

2(b)(iii)

1. Find the number of students who listen to classical music only.

From part (ii),
$n(A\cap B)=10.$
Classical music only:
$n(B)-n(A\cap B)=14-10=4.$

答案

$4$

2(b)(iv)

1. Subtract the number of students in at least one of the two sets from the total number of students.

Number who listen to at least one of pop or classical music
$=12+10+4=26$

Number who listen to neither
$=35-26=9$

答案

$9$

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