Solution Found!
a. A lumber company has just taken delivery on a lot of
Chapter 3, Problem 86E(choose chapter or problem)
a. A lumber company has just taken delivery on a lot of 10, 002 × 4 boards. Suppose that 20% of these boards (2,000) 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 lot 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) ?
Questions & Answers
QUESTION:
a. A lumber company has just taken delivery on a lot of 10, 002 × 4 boards. Suppose that 20% of these boards (2,000) 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 lot 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) ?
ANSWER:Solution : Step 1: a) It is given that the company taken delivery on a lot of 10,000 boards. And 20% of the these boards(2000) are too green to be used in first quality construction. We selected two boards randomly. we have to find P(A),P(B) and P(A B ). Since it is given A = { the first board is green} and B = { the second board is green} By using the definition of probability we can calculate the probability that the first board selected as green N(A) number of green boards 2000 P(A)= N = total number of boards = 10000 =0.2 Since there is no replacement in this experiment, the probability that the second board is green depends on whether or not the first board selected was green. So it is clear that A and B are not independent. Let A {the first board is not green} So P(A =1-P(A)= 1-0.2 = 0.8 Here the two possible ways that B can