a. A lumber company has just taken delivery on a shipment of 10,000 2 × 4 boards. Suppose that 20% of these boards (2000) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let A = {the first board is green} and B = {the second board is green}. Compute P(A), P(B), and P(A ⋂ B) (a tree diagram might help). Are A and B independent?
b. With A and B independent and P(A) = P(B) = .2, what is P(A ⋂ B)? How much difference is there between this answer and P(A ⋂ B) in part (a)? For purposes of calculating P(A ⋂ B), can we assume that A and B of part (a) are independent to obtain essentially the correct probability?
c. Suppose the shipment consists of ten boards, of which two are green. Does the assumption of independence now yield approximately the correct answer for P(A ⋂ B)? What is the critical difference between the situation here and that of part (a)? When do you think an independence assumption would be valid in obtaining an approximately correct answer to P(A ⋂ B)?