To determine whether the events [tex]\(A\)[/tex] (selecting a plastic bottle) and [tex]\(B\)[/tex] (the item was recycled by a tenth grader) are independent, we need to check if [tex]\(P(A \mid B) = P(A)\)[/tex]. Here’s how we can proceed step by step:
### Step 1: Calculate [tex]\(P(A)\)[/tex]
[tex]\(P(A)\)[/tex] is the probability of selecting a plastic bottle from the entire set of recycled items.
From the table:
- Total number of recycled items [tex]\(= 400\)[/tex]
- Number of plastic bottles [tex]\(= 135\)[/tex]
[tex]\[P(A) = \frac{\text{Number of plastic bottles}}{\text{Total number of recycled items}} = \frac{135}{400}\][/tex]
### Step 2: Calculate [tex]\(P(A \mid B)\)[/tex]
[tex]\(P(A \mid B)\)[/tex] is the probability of selecting a plastic bottle given that it was recycled by a tenth grader.
From the table:
- Total number of items recycled by tenth graders [tex]\(= 150\)[/tex]
- Number of plastic bottles recycled by tenth graders [tex]\(= 40\)[/tex]
[tex]\[P(A \mid B) = \frac{\text{Number of plastic bottles recycled by tenth graders}}{\text{Total number of items recycled by tenth graders}} = \frac{40}{150}\][/tex]
### Step 3: Compare [tex]\(P(A)\)[/tex] and [tex]\(P(A \mid B)\)[/tex]
Now, we compare the two probabilities:
- [tex]\(P(A) = \frac{135}{400}\)[/tex]
- [tex]\(P(A \mid B) = \frac{40}{150}\)[/tex]
We notice from the given information that [tex]\(P(A) \neq P(A \mid B)\)[/tex].
### Conclusion
Since [tex]\(P(A) \neq P(A \mid B)\)[/tex], the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are not independent. Therefore, the correct statement is:
[tex]$A$[/tex] and [tex]$B$[/tex] are not independent events because [tex]\(P(A \mid B) \neq P(A)\)[/tex].
So, the answer is:
[tex]\[ \text{A and B are not independent events because } P(A \mid B) \neq P(A). \][/tex]
