Question

information about the recycling drive at school is shown in the table. let a be the event that the item pulled out of the recycling bin is a plastic bottle, and let b be the event that a tenth grader recycled that item.

which statement is true about whether a and b are independent events?

• a and b are independent events because \\(p(a|b) = p(a)\\)
• a and b are independent events because \\(p(a|b) = p(b)\\)
• a and b are not independent events because \\(p(a|b) \

eq p(a)\\)

• a and b are not independent events because \\(p(a|b) \

eq p(b)\\)

Explanation:

Define the events and extract values

Using the Two-Way Frequency Tables knowledge point

• Total number of items: \(N = 400\)
• Event \(A\): The item is a plastic bottle.
• Total plastic bottles: \(135\)
• Event \(B\): A tenth grader recycled the item.
• Total items recycled by tenth graders: \(150\)
• Intersection \(A \cap B\): Plastic bottles recycled by tenth graders: \(40\)

Calculate the probability of event A

Using the Conditional Probability Calculation knowledge point

$$ P(A) = \frac{\text{Total Plastic Bottles}}{\text{Grand Total}} = \frac{135}{400} = 0.3375 $$

Calculate the conditional probability P(A|B)

Using the Conditional Probability Calculation knowledge point

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{40}{150} \approx 0.2667 $$

Determine independence

Using the Independent Events knowledge point

$$
LATEXBLOCK0
$$

Since the conditional probability of \(A\) given \(B\) is not equal to the unconditional probability of \(A\), the events are not independent.

Answer:

• (A) A and B are independent events because \(P(A|B) = P(A)\)
• (B) A and B are independent events because \(P(A|B) = P(B)\)</mcq-correct>
• **(C) A and B are not independent events because \(P(A|B)

eq P(A)\) (Correct answer)**
<mcq-option>(D) A and B are not independent events because \(P(A|B)
eq P(B)\)