To determine if being a girl and playing a sport are independent events, we need to compare the probability of being a girl, [tex]\( P(\text{girl}) \)[/tex], with the conditional probability of being a girl given that the student plays a sport, [tex]\( P(\text{girl} \mid \text{plays a sport}) \)[/tex].
Here is our step-by-step approach:
1. Calculate the total number of students, total number of girls, and total number of students playing a sport:
[tex]\[
\text{Total students} = 275
\][/tex]
[tex]\[
\text{Total girls} = 135
\][/tex]
[tex]\[
\text{Total students playing a sport} = 171
\][/tex]
[tex]\[
\text{Girls playing a sport} = 76
\][/tex]
2. Calculate the probability of a student being a girl, [tex]\( P(\text{girl}) \)[/tex]:
[tex]\[
P(\text{girl}) = \frac{\text{Total girls}}{\text{Total students}} = \frac{135}{275} \approx 0.4909
\][/tex]
3. Calculate the conditional probability of being a girl given that the student plays a sport, [tex]\( P(\text{girl} \mid \text{plays a sport}) \)[/tex]:
[tex]\[
P(\text{girl} \mid \text{plays a sport}) = \frac{\text{Girls playing a sport}}{\text{Total students playing a sport}} = \frac{76}{171} \approx 0.4444
\][/tex]
4. Determine if the events are independent:
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if [tex]\( P(A \mid B) = P(A) \)[/tex]. In this case, we need to check if [tex]\( P(\text{girl} \mid \text{plays a sport}) = P(\text{girl}) \)[/tex].
Comparing the values we have:
[tex]\[
P(\text{girl}) \approx 0.4909
\][/tex]
[tex]\[
P(\text{girl} \mid \text{plays a sport}) \approx 0.4444
\][/tex]
Since [tex]\( 0.4909 \neq 0.4444 \)[/tex], the two probabilities are not equal. Hence, being a girl and playing a sport are not independent events.
Therefore, the correct answer is:
[tex]\[ B. \text{No, they are not independent, because } P(\text{girl}) \approx 0.49 \text{ and } P(\text{girl} \mid \text{plays a sport}) \approx 0.44. \][/tex]
