Let's break down the answers step by step:
1. Calculating [tex]\(P(\text{Boy})\)[/tex]:
- The probability of selecting a boy from the total number of students.
- Total number of boys = 160
- Total number of students = 240
[tex]\[
P(\text{Boy}) = \frac{\text{Number of Boys}}{\text{Total Students}} = \frac{160}{240} = \frac{2}{3} \approx 0.67
\][/tex]
2. Calculating [tex]\(P(\text{Boy} \mid \text{Indoor Recess})\)[/tex]:
- The probability of selecting a boy given that the student prefers indoor recess.
- Number of boys who prefer indoor recess = 64
- Total number of students who prefer indoor recess = 96
[tex]\[
P(\text{Boy} \mid \text{Indoor Recess}) = \frac{\text{Number of Boys who prefer Indoor Recess}}{\text{Total Indoor Recess Students}} = \frac{64}{96} = \frac{2}{3} \approx 0.67
\][/tex]
3. Determining if the events are independent:
- Events are independent if:
[tex]\[
P(\text{Boy and Indoor Recess}) = P(\text{Boy}) \times P(\text{Indoor Recess})
\][/tex]
- Calculate [tex]\(P(\text{Boy and Indoor Recess})\)[/tex]:
- Number of boys who prefer indoor recess = 64
- Total number of students = 240
[tex]\[
P(\text{Boy and Indoor Recess}) = \frac{64}{240} = \frac{4}{15}
\][/tex]
- Calculate [tex]\(P(\text{Boy}) \times P(\text{Indoor Recess})\)[/tex]:
- [tex]\(P(\text{Indoor Recess})\)[/tex]:
- Total number of students who prefer indoor recess = 96
- Total number of students = 240
[tex]\[
P(\text{Indoor Recess}) = \frac{96}{240} = \frac{2}{5}
\][/tex]
[tex]\[
P(\text{Boy}) \times P(\text{Indoor Recess}) = \left( \frac{2}{3} \right) \times \left( \frac{2}{5} \right) = \frac{4}{15}
\][/tex]
- Since:
[tex]\[
P(\text{Boy and Indoor Recess}) = P(\text{Boy}) \times P(\text{Indoor Recess}) = \frac{4}{15}
\][/tex]
- The events are independent since the probabilities match.
Therefore, your final answers in the drop-down menu should be:
[tex]\[
\begin{array}{l}
P(\text { Boy })= \frac{2}{3} \\
P(\text { Boy } \mid \text { Indoor Recess })= \frac{2}{3}
\end{array}
\][/tex]
[tex]\[
\text{The events of the student being a boy and the student preferring indoor recess are \textbf{independent}}
\][/tex]
