To determine whether being a student and preferring sharks are independent events, we need to compare the probability of being a student ([tex]\(P(\text{student})\)[/tex]) with the conditional probability of being a student given that the person prefers sharks ([tex]\(P(\text{student | shark})\)[/tex]).
1. Calculate [tex]\(P(\text{student})\)[/tex]:
[tex]\[
P(\text{student}) = \frac{\text{Number of students}}{\text{Total number of people surveyed}} = \frac{100}{115} \approx 0.87
\][/tex]
2. Calculate [tex]\(P(\text{student | shark})\)[/tex]:
From the table, there are 90 students who prefer sharks out of the total 95 people who prefer sharks.
[tex]\[
P(\text{student | shark}) = \frac{\text{Number of students who prefer sharks}}{\text{Total number of people who prefer sharks}} = \frac{90}{95} \approx 0.95
\][/tex]
3. Compare [tex]\(P(\text{student})\)[/tex] and [tex]\(P(\text{student | shark})\)[/tex]:
[tex]\[
P(\text{student}) \approx 0.87
\][/tex]
[tex]\[
P(\text{student | shark}) \approx 0.95
\][/tex]
Since [tex]\(P(\text{student})\)[/tex] is not equal to [tex]\(P(\text{student | shark})\)[/tex], the events are not independent.
Therefore, the correct answer is:
D. No, they are not independent because [tex]\(P(\text{student}) \approx 0.87\)[/tex] and [tex]\(P(\text{student | shark}) \approx 0.95\)[/tex].
