Question

a magazine surveyed subscribers, asking which of the following reality shows they watched on a regular basis: survivor, the voice, americas got talent. the results showed that 358 watched survivor, 295 watched the voice, 285 watched americas got talent, 194 watched survivor and the voice, 202 watched survivor and americas got talent, 139 watched the voice and americas got talent, 68 watched survivor, the voice, and americas got talent, and 27 watched none of these shows. complete parts a) through e) below.
a) how many subscribers were surveyed?
498 (simplify your answer.)
b) of the subscribers surveyed, how many watched survivor and americas got talent, but not the voice?
134 (simplify your answer.)
c) of the subscribers surveyed, how many watched the voice, but neither survivor nor americas got talent?
(simplify your answer.)

Explanation:

Step1: Define the formula for three - set inclusion - exclusion

For three sets \(A\) (Survivor), \(B\) (The Voice), and \(C\) (America's Got Talent), the number of elements in \(B\) but not in \(A\) or \(C\) is given by \(n(B)-n(A\cap B)-n(B\cap C)+n(A\cap B\cap C)\). Here, \(n(B) = 295\) (number of people who watched The Voice), \(n(A\cap B)=194\) (number of people who watched Survivor and The Voice), \(n(B\cap C) = 139\) (number of people who watched The Voice and America's Got Talent), and \(n(A\cap B\cap C)=68\) (number of people who watched all three shows).

Step2: Substitute the values into the formula

\[

$$\begin{align*} &n(B)-n(A\cap B)-n(B\cap C)+n(A\cap B\cap C)\\ =&295 - 194-139 + 68\\ =&(295+68)-(194 + 139)\\ =&363-333\\ =&30 \end{align*}$$

\]

Answer:

\(30\)