To determine whether being a student and preferring "Viking" are independent events, we'll need to look at the probabilities involved. Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if [tex]\(P(A \mid B) = P(A)\)[/tex].
Given the survey data:
[tex]\[
\begin{tabular}{|l|c|c|c|}
\hline & Vikings & Patriots & Total \\
\hline Students & 80 & 20 & 100 \\
\hline Teachers & 5 & 20 & 25 \\
\hline Total & 85 & 40 & 125 \\
\hline
\end{tabular}
\][/tex]
1. Calculate [tex]\(P(\text{student})\)[/tex]:
The probability of selecting a student, [tex]\(P(\text{student})\)[/tex], is the number of students divided by the total number of people surveyed.
[tex]\[
P(\text{student}) = \frac{\text{Total number of students}}{\text{Total number of people}} = \frac{100}{125} = 0.8
\][/tex]
2. Calculate [tex]\(P(\text{student} \mid \text{Viking})\)[/tex]:
The conditional probability of being a student given that the person prefers "Viking" is the number of students who prefer "Viking" divided by the total number of people who prefer "Viking".
[tex]\[
P(\text{student} \mid \text{Viking}) = \frac{\text{Number of students who prefer Viking}}{\text{Total number of people who prefer Viking}} = \frac{80}{85} \approx 0.941176
\][/tex]
3. Compare [tex]\(P(\text{student})\)[/tex] and [tex]\(P(\text{student} \mid \text{Viking})\)[/tex]:
- [tex]\(P(\text{student}) = 0.8\)[/tex]
- [tex]\(P(\text{student} \mid \text{Viking}) \approx 0.941176\)[/tex]
Since [tex]\(P(\text{student})\)[/tex] is not equal to [tex]\(P(\text{student} \mid \text{Viking})\)[/tex], the events "being a student" and "preferring Viking" are not independent.
Therefore, the correct answer is:
A. No, they are not independent because [tex]\(P(\text{student}) = 0.80\)[/tex] and [tex]\(P(\text{student} \mid \text{Viking}) \approx 0.94\)[/tex].
