Answered

Select the correct answer from each drop-down menu.

A survey asking about preference for recess location was randomly given to students in an elementary school. The results are recorded in the table below.

[tex]\[
\begin{tabular}{|l|l|l|l|}
\hline & Indoor Recess & Outdoor Recess & Total \\
\hline Boys & 64 & 96 & 160 \\
\hline Girls & 32 & 48 & 80 \\
\hline Total & 96 & 144 & 240 \\
\hline
\end{tabular}
\][/tex]

A student is randomly selected. Based on the data, what conclusions can be drawn?

[tex]\[
\begin{array}{l}
P(\text{Boy}) = \square \\
P(\text{Boy} \mid \text{Indoor Recess}) = \square
\end{array}
\][/tex]

The events of the student being a boy and the student preferring indoor recess are [tex]$\square$[/tex].



Answer :

GinnyAnswer
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]