Question

In a survey of $120$ people, it was found that $50$ people read newspaper $H,52$ people read newspaper $T,52$ people read newspaper $I,18$ read both $H$ and $I,22$ read both $H$ and $T,16$ read both $T$ and $I,6$ read all there news papers. Find:
i) the number of people who read exactly one newspaper.
ii) the number of people who read exactly two news papers.
iii) the number of people who read at least one of the newspapers.

Answer 1

We have $S=120$

i) The number of people who read exactly one newspaper

$\Rightarrow P\left(H\right)+P\left(T\right)+P\left(I\right)-2\left(\right(H\cap T)+P(T\cap I)+P(I\cap H\left)\right)+3P(H\cap T\cap I)=50+52+52-2(18+22+16)+3\left(6\right)=60$

ii) The number of people who read exactly two newspapers

$\Rightarrow \left(\right(H\cap T)+P(T\cap I)+P(I\cap H\left)\right)-3P(H\cap T\cap I)=(18+22+16)-3\left(6\right)=38$

iii) the number of people who read at least one of the newspapers

$\Rightarrow P(H\cup T\cup I\cup )=P\left(H\right)+P\left(T\right)+P\left(I\right)-\left(\right(H\cap T)+P(T\cap I)+P(I\cap H\left)\right)+P(H\cap T\cap I)=50+52+52-(18+22+16)+\left(6\right)=104$