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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.
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:
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