Answer
Five hundred respondents said “yes” to the taxes question but “no” to the homeland security question.
Work Step by Step
Let A be the event that the respondents said “yes” to the taxes question.
Let B be the event that the respondents said “yes” to the homeland security question.
We have given that,
\[N(S)=1200\]
\[N(A)=600\]
\[N(B)=400\]
\[N({{A}^{C}}\cap B)=300\]
\[\begin{align}
& N({{B}^{C}})=N(S)-N(B) \\
& N({{B}^{C}})=1200-400 \\
& N({{B}^{C}})=800 \\
\end{align}\]
We have to find the number of respondents said “yes” to the taxes question but “no” to the homeland security question; that means\[N(A\cap {{B}^{C}})\].
We can write \[N(A\cap {{B}^{C}})\]as:
\[N(A\cap {{B}^{C}})=N(A)+N({{B}^{C}})-N(A\cup {{B}^{C}})\]
We first find \[N(A\cup {{B}^{C}})\],
\[\begin{align}
& N(A\cup {{B}^{C}})=N(S)-N({{A}^{C}}\cap B) \\
& N(A\cup {{B}^{C}})=1200-300 \\
& N(A\cup {{B}^{C}})=900 \\
\end{align}\]
Now we get,
\[\begin{align}
& N(A\cap {{B}^{C}})=N(A)+N({{B}^{C}})-N(A\cup {{B}^{C}}) \\
& N(A\cap {{B}^{C}})=600+800-900 \\
& N(A\cap {{B}^{C}})=500 \\
\end{align}\]
Therefore, five hundred respondents said “yes” to the taxes question but “no” to the homeland security question.
