To determine whether the events A (a student likes apples) and B (a student is in middle school) are independent, we need to check if the following condition holds true:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Where:
- \(P(A \cap B)\) is the probability that a student likes apples and is in middle school.
- \(P(A)\) is the probability that a student likes apples.
- \(P(B)\) is the probability that a student is in middle school.
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Calculate \(P(A)\): The total number of students is 300. The number of students who like apples is 274. \[ P(A) = \frac{274}{300} = \frac{137}{150} \]
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Calculate \(P(B)\): The total number of middle school students is 100. \[ P(B) = \frac{100}{300} = \frac{1}{3} \]
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Calculate \(P(A \cap B)\): The number of middle school students who like apples is 92. \[ P(A \cap B) = \frac{92}{300} \]
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Now, calculate \(P(A) \cdot P(B)\): \[ P(A) \cdot P(B) = \left(\frac{274}{300}\right) \cdot \left(\frac{100}{300}\right) = \frac{274 \times 100}{300 \times 300} = \frac{27400}{90000} = \frac{274}{900} \]
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Compare \(P(A \cap B)\) and \(P(A) \cdot P(B)\): \[ P(A \cap B) = \frac{92}{300} = \frac{92}{300} \cdot \frac{3}{3} = \frac{276}{900} \]
Now, we can see:
- \(P(A \cap B) = \frac{276}{900}\)
- \(P(A) \cdot P(B) = \frac{274}{900}\)
Since \(P(A \cap B) \neq P(A) \cdot P(B)\), the two events A and B are not independent.
Conclusion: Events A (likes apples) and B (is in middle school) are not independent.