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.

A 5-column table has 4 rows. The first column has entries tenth grade, eleventh grade, twelfth grade, total. The second column is labeled aluminum cans with entries 80, 56, 64, 200. The third column is labeled glass bottles with entries 30, 9, 26, 65. The fourth column is labeled plastic bottles with entries 40, 45, 50, 135. The fifth column is labeled Total with entries 150, 110, 140, 400.

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) ≠ P(A).
A and B are not independent events because P(A∣B) ≠ P(B).

Answer 1

Compute the probabilities:
- P(A) = plastic = 135/400 = 0.3375.
- P(B) = tenth grader = 150/400 = 0.375.
- P(A|B) = plastic given tenth = 40/150 ≈ 0.2667.

Since P(A|B) ≠ P(A), A and B are not independent. Therefore the correct statement is:
"A and B are not independent events because P(A∣B) ≠ P(A)."