Question

Apple Preference/Elementary School Students/Middle School Students/High School Students/Total
Likes apples/85/92/97/274
Dislikes apples/15/8/3/26
Total/100/100/100/300

Let A be the event that a student likes apples. Let B be the event that a student is in middle school. Based on the data in the table, are events A and B independent?

Answer 1

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.

1. Calculate \(P(A)\):
The total number of students is 300. The number of students who like apples is 274.
\[
P(A) = \frac{274}{300} = \frac{137}{150}
\]

2. Calculate \(P(B)\):
The total number of middle school students is 100.
\[
P(B) = \frac{100}{300} = \frac{1}{3}
\]

3. Calculate \(P(A \cap B)\):
The number of middle school students who like apples is 92.
\[
P(A \cap B) = \frac{92}{300}
\]

4. Now, calculate \(P(A) \cdot P(B)\):
\[
P(A) \cdot P(B) = \left(\frac{274}{300}\right) \cdot \left(\frac{100}{300}\right) = \frac{274 \times 100}{300 \times 300} = \frac{27400}{90000} = \frac{274}{900}
\]

5. 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.