Question

1. a local television station sends out questionnaires to determine if viewers would rather see a documentary, an interview show, or reruns of a game show. there were 600 responses with the following results: 180 were interested in an interview show and a documentary, but not reruns. 24 were interested in an interview show and reruns but not a documentary. 84 were interested in reruns but not an interview show. 144 were interested in an interview show but not a documentary. 60 were interested in a documentary and reruns. 36 were interested in an interview show and reruns. 48 were interested in none of the three. how many are interested in exactly one kind of show?

Explanation:

Step1: Define sets

Let \(A\) be the set of viewers interested in an interview - show, \(B\) be the set of viewers interested in a documentary, and \(C\) be the set of viewers interested in reruns of a game - show. The total number of responses is \(n = 600\), and the number of viewers interested in none of the three is \(48\). So the number of viewers interested in at least one of the shows is \(n'=600 - 48=552\).

Step2: Analyze given data

We know:

• Number interested in \(A\cap B\cap\overline{C}=180\)
• Number interested in \(A\cap\overline{B}\cap C = 24\)
• Number interested in \(\overline{A}\cap\overline{B}\cap C=84\)
• Number interested in \(A\cap\overline{B}\cap\overline{C}=144 - 24=120\) (since \(144\) is interested in \(A\) but not \(B\), and \(24\) of them are also interested in \(C\))
• Number interested in \(\overline{A}\cap B\cap C = 60\)
• Number interested in \(A\cap B\cap C=36 - 24=12\) (since \(36\) are interested in \(A\) and \(C\), and \(24\) of them are not interested in \(B\))

Step3: Calculate number interested in exactly one show

Number interested in only \(A\): \(n(A\cap\overline{B}\cap\overline{C}) = 120\)
Number interested in only \(B\):
We first find the number of elements in \(B\) using the principle of inclusion - exclusion. But to find only \(B\), we note that \(n'=n(A\cup B\cup C)\).
The number interested in only \(B\): \(n(\overline{A}\cap B\cap\overline{C})\)
We know \(n'=n(A\cap\overline{B}\cap\overline{C})+n(\overline{A}\cap B\cap\overline{C})+n(\overline{A}\cap\overline{B}\cap C)+n(A\cap B\cap\overline{C})+n(A\cap\overline{B}\cap C)+n(\overline{A}\cap B\cap C)+n(A\cap B\cap C)\)
We can find \(n(\overline{A}\cap B\cap\overline{C})\) as follows:
\[

$$\begin{align*} n(\overline{A}\cap B\cap\overline{C})&=n'-n(A\cap\overline{B}\cap\overline{C})-n(\overline{A}\cap\overline{B}\cap C)-n(A\cap B\cap\overline{C})-n(A\cap\overline{B}\cap C)-n(\overline{A}\cap B\cap C)-n(A\cap B\cap C)\\ &=552-(120 + 84+180 + 24+60 + 12)\\ &=552 - 480\\ &=72 \end{align*}$$

\]
The number interested in only \(C\): \(n(\overline{A}\cap\overline{B}\cap C)=84\)
The number interested in exactly one show is \(120 + 72+84=276\)

Answer:

276