Question

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I,11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find: (i) the number of people who read at least one of the newspapers. (ii) the number of people who read exactly one newspaper.

Answer 1

(i)

The total number of people in the survey is 60 .

Let, A , B and C be the number of people who read newspapers H, T and I respectively.

Therefore, n( A )=25

n( B )=26

n( C )=26

n( A∩C )=9

n( A∩B )=11

The total number of people who read both newspapers T and I is 8 .

n( B∩C )=8

n( A∩B∩C )=3

From the formula of union of sets A , B and C is,

n( A∪B∪C )=n( A )+n( B )+n( C )−n( A∩B ) −n( A∩C )−n( B∩C )+n( A∩B∩C ) n( A∪B∪C )=25+26+26−11−9−8+3 =52

Thus, the number of people who read at least one of the newspapers is 52 .

(ii)

Let, f denote the number of people read newspaper H, g denote the number of people read newspaper T and h denote the number of people read newspaper I.

Let, a be the number of people who read newspaper H and T only, b be the number of people who read newspaper I and H only, c be the number of people who read newspaper T and I only, d be the number of people who read all the newspapers.

According to given information,

n( A∩B∩C )=d=3

Now,

n( A∩B )=11=a+d n( B∩C )=8=c+d n( C∩A )=9=b+d

Therefore,

n( A∩B )+n( B∩C )+n( C∩A )=a+d+c+d+b+d 11+8+9=a+c+b+d+2d a+c+b+d=28−2( 3 ) a+c+b+d=22

The number of people who read at least one of the newspapers is n( A∪B∪C )=52

The number of people who read exactly one newspaper is,

52−22=30

Thus, the number of people who read exactly one newspaper is 30 .