To determine whether two events are independent, we need to compare the probability of one event happening alone with the probability of that event happening given that the other event has occurred. In this case, we need to compare the probability of being a student ([tex]\( P(\text{student}) \)[/tex]) with the probability of being a student given that the person prefers sharks ([tex]\( P(\text{student} \mid \text{shark}) \)[/tex]).
### Step-by-Step Solution
1. Calculate [tex]\( P(\text{student}) \)[/tex]:
- Total number of people surveyed = 115
- Total number of students = 100
[tex]\[
P(\text{student}) = \frac{\text{Total number of students}}{\text{Total number of people surveyed}} = \frac{100}{115} \approx 0.8696
\][/tex]
2. Calculate [tex]\( P(\text{student} \mid \text{shark}) \)[/tex]:
- Total number of people preferring sharks = 95
- Number of students preferring sharks = 90
[tex]\[
P(\text{student} \mid \text{shark}) = \frac{\text{Number of students preferring sharks}}{\text{Total number of people preferring sharks}} = \frac{90}{95} \approx 0.9474
\][/tex]
3. Compare the probabilities:
- [tex]\( P(\text{student}) \approx 0.8696 \)[/tex]
- [tex]\( P(\text{student} \mid \text{shark}) \approx 0.9474 \)[/tex]
Since [tex]\( P(\text{student}) \)[/tex] and [tex]\( P(\text{student} \mid \text{shark}) \)[/tex] are not approximately equal (0.8696 is not close to 0.9474), the events are not independent.
### Conclusion
Based on the calculations, being a student and preferring sharks are not independent events. Therefore, the correct answer is:
B. No, they are not independent because [tex]\( P(\text{student}) \approx 0.87 \)[/tex] and [tex]\( P(\text{student} \mid \text{shark}) \approx 0.95 \)[/tex].
