To determine whether being a student and preferring "pirate" are independent events, we need to compare the probability of a person being a student, [tex]\(P(\text{student})\)[/tex], with the conditional probability of a person being a student given that they prefer pirates, [tex]\(P(\text{student} \mid \text{pirate})\)[/tex].
### Step-by-Step Calculation:
1. Total number of people surveyed:
[tex]\[
110
\][/tex]
2. Total number of students:
[tex]\[
90
\][/tex]
3. Total number of people who prefer pirates:
[tex]\[
80
\][/tex]
4. Number of students who prefer pirates:
[tex]\[
75
\][/tex]
5. Probability of a person being a student:
[tex]\[
P(\text{student}) = \frac{\text{Total number of students}}{\text{Total number of people surveyed}} = \frac{90}{110} \approx 0.818
\][/tex]
6. Probability of a person preferring pirates:
[tex]\[
P(\text{pirate}) = \frac{\text{Total number of people who prefer pirates}}{\text{Total number of people surveyed}} = \frac{80}{110} \approx 0.727
\][/tex]
7. Conditional probability of a person being a student given they prefer pirates:
[tex]\[
P(\text{student} \mid \text{pirate}) = \frac{\text{Number of students who prefer pirates}}{\text{Total number of people who prefer pirates}} = \frac{75}{80} = 0.9375
\][/tex]
### Analysis of Independence:
To check for independence, we compare [tex]\(P(\text{student})\)[/tex] and [tex]\(P(\text{student} \mid \text{pirate})\)[/tex]. If the events are independent, these probabilities should be equal.
We have:
[tex]\[
P(\text{student}) \approx 0.818
\][/tex]
[tex]\[
P(\text{student} \mid \text{pirate}) = 0.9375
\][/tex]
Since [tex]\(P(\text{student}) \approx 0.818\)[/tex] and [tex]\(P(\text{student} \mid \text{pirate}) = 0.9375\)[/tex] are not equal, the events are not independent.
### Conclusion:
The correct answer is:
C. No, they are not independent because [tex]\(P(\text{student}) \approx 0.82\)[/tex] and [tex]\(P(\text{student} \mid \text{pirate}) \approx 0.94\)[/tex].
