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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 numbers of people who read at least one of the newspapers. (ii) the number of people who read exactly one newspaper.

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Solution

## $\mathrm{Given}:\phantom{\rule{0ex}{0ex}}n\left(H\right)=25\phantom{\rule{0ex}{0ex}}n\left(T\right)=26\phantom{\rule{0ex}{0ex}}n\left(I\right)=26\phantom{\rule{0ex}{0ex}}n\left(H\cap I\right)=9\phantom{\rule{0ex}{0ex}}n\left(H\cap T\right)=11\phantom{\rule{0ex}{0ex}}n\left(T\cap I\right)=8\phantom{\rule{0ex}{0ex}}n\left(H\cap T\cap I\right)=3\phantom{\rule{0ex}{0ex}}\left(\mathrm{i}\right)\mathrm{We}\mathrm{know}:\phantom{\rule{0ex}{0ex}}n\left(H\cup T\cup I\right)=n\left(H\right)+n\left(T\right)+n\left(I\right)-n\left(H\cap T\right)-n\left(T\cap I\right)-n\left(H\cap I\right)+n\left(H\cap T\cap I\right)\phantom{\rule{0ex}{0ex}}⇒n\left(H\cup T\cup I\right)=25+26+26-11-8-9+3=52\phantom{\rule{0ex}{0ex}}\mathrm{Thus},52\mathrm{people}\mathrm{can}\mathrm{read}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{newspapers}.\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right)\mathrm{Now},\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{calculate}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{people}\mathrm{who}\mathrm{read}\mathrm{exactly}\mathrm{one}\mathrm{newspaper}.\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{have}:\phantom{\rule{0ex}{0ex}}n\left(H\right)+n\left(T\right)+n\left(I\right)-2n\left(H\cap T\right)-2n\left(T\cap I\right)-2n\left(H\cap I\right)+3n\left(H\cap T\cap I\right)\phantom{\rule{0ex}{0ex}}=25+26+26-22-16-18+9\phantom{\rule{0ex}{0ex}}=30\phantom{\rule{0ex}{0ex}}\mathrm{Thus},30\mathrm{people}\mathrm{can}\mathrm{read}\mathrm{exactly}\mathrm{one}\mathrm{newspaper}.$

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