In a particular month, 21 people took a driving test. One of the 21 people is selected at random. The probability that it is a man who passed the test is $\frac{1}{7}$. Two of the 21 people are selected at random. The probability that they are both women who failed the test is $\frac{1}{10}$. Complete the table of information about the 21 people who took the test on that particular month. (b) $\varepsilon = \{x:\ x\text{ is a student in a class}\}$ $A = \{x:\ x\text{ is a student who listens to pop music}\}$ $B = \{x:\ x\text{ is a student who listens to classical music}\}$ It is given that $n(\varepsilon)=35$, $n(A)=22$, $n(B)=14$ and $n(A\cap B')=12$. Describe in words the set $A\cap B'$. ... Find the number of students who listen to both pop and classical music. Answer ... Find the number of students who listen to classical music only. Answer ... Find the number of students who listen to neither pop nor classical music. Answer ...

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
$\text{women passed}=21-3-9-7$
$=2$

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 $B'$.

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

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

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

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

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
$=35-26=9$