A survey of television viewers at a Child's Place Preschool produces the following data.

Question

A survey of television viewers at a Child's Place Preschool produces the following data:

60% watch Sesame Street. 50% watch Captain Kangaroo. 50% watch Polka Dot Door. 30% watch Sesame Street and Captain Kangaroo. 20% watch Captain Kangaroo and Polka Dot Door. 30% watch Sesame Street and Polka Dot Door. 10% watch all three shows.

What percentage views at least one of these programs? What percentage views none of these shows?

NOTE: I'm expected to use the Principle of Inclusion and Exclusion here, but I don't understand how you can do that with percentages. I tried giving a pretend population for the survey of television viewers, say 100 people, so that I could write convert percentages to instead, "60 people watch Sesame Street, 50 people watch Captain Kangaroo...", but that didn't seem to work.

Could someone help me out? Thanks.

What I did:

Answer 1

You're on the right track; you just need to get the three outer areas right. You've got 50 people already watching Sesame Street; since there are 60 total, there must be 10 who watch Sesame Street and nothing else.

You should be able to do something similar to get the number who watch Captain Kangaroo or Polka Dot Door and nothing else; then, add up all the numbers in all the different regions and you're golden.

Answer 2

For the Venn diagram, let's adopt your idea of using $100$ people. You correctly labeled the number of people who watch all three shows and the number of people who watch exactly two shows. We know that $50\%$ of the viewers watch Captain Kangaroo. Thus far, you have $20 + 10 + 10 = 40$ people watching Captain Kangaroo and at least one other show. Thus, $50 - 40 = 10$ watch only Captain Kangaroo. By similar reasoning, exactly $10$ people watch only Polka Dot Door and exactly $10$ people watch only Sesame Street. If we add the totals in these disjoint subsets together, we find that $90$ people watch at least one of the shows, leaving $100 - 90 = 10$ people who watch none of them, as the Venn diagram below shows.

If we use the Inclusion-Exclusion Principle, the percentage of viewers who watch at least one of the shows is \begin{align*} |C \cup P \cup S| & = |C| + |P| + |S| - |C \cap P| - |C \cap S| - |P \cap S| + |C \cap P \cap S|\\ & = 50\% + 50\% + 60\% - 20\% - 30\% - 30\% + 10\%\\ & = 90\% \end{align*} from which we can conclude that $100\% - 90\% = 10\%$ watch none of the three shows.