A lumber company has just taken delivery on a shipment of 10,000 2 times 4 board

Question

A lumber company has just taken delivery on a shipment of 10,000 2 times 4 boards. Suppose that 40% of these boards (4000) 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 he second board is green). a) Compute P(A), P(B) and P(A intersection B) (a tree diagram might help). Round your answer for P(A intersection B) to five decimal places.) Are A and B independent Yes, the two events are independent. No, the two events are not independent. b) With A and independent and P(A) = P(B) = 0, 4, what is P(A intersection B)? How much difference is there between this answer and P(A intersection B) in part (a)? There is no difference. There is very little difference. There is a very large difference. For purposes of calculating P(A intersection B) can we assume that A and B of pant (a) are independent to obtain essentially the correct probability? Yes No (c) Suppose the lot consists of ten boards, of which four are green. Does the assumption of independence now yield approximately the correct answer for P(A intersection B)? Yes No What is the critical difference between the situation here and that of part (a) The critical difference is that the percentage of green boards is smaller in part (a). The critical difference is that the population size in part (a) is small compared to the random sample of two boards. The critical difference is that the population size in part (a) is huge compared to the random sample of two boards. The critical difference is that the percentage of green boards is larger in part (a). When do you think that an independence assumption would be valid in obtaining an approximately correct answer to P(A intersection B)? This assumption would be valid when the population is much larger than the sample size. This assumption would be valid when there are more green boards in the sample. This assumption would be valid when there are fewer green boards in the sample. This assumption would be valid when the sample size is very large.

Explanation / Answer

A ANS)

A = {the first board is green} B = {the second board is green}

Using the definition of probability, we may calculate the probability that the first board selected is green as

P(A) = N(A) / n = 4000/1000 = 0.4

According to the definition of complementary events, P(A¢) = 1 – P(A) = 1 – 0.4 = 0.6

The two possible ways that event B may occur are

P(B / A) = N(B/A) / N-1 = 3999 / 9999=0.399

P(B/A') =  N(B/A') / N-1 = 4000/9999=0.400

The probability that the second board selected is green may be found by applying the total

P(B) = P(B | A) ×P(A) + P(B | A¢) ×P(A¢)= 0.399*0.4+0.4*0.6 =0.3996

The probability that the first two boards selected are green may be found using the multiplication rule.

P(A B) = P(B | A) ×B = 0.399*0.4=0.1596

ARE A AND B ARE INDEPENDENT ?

no they are not independent We may verify this by establishing the following inequality, which is a violation of the definition of independence. P(A  B) = 0.1596¹ not equal to 0.04 = P(A) ×P(B)

b) p(a) = p(b)= 0.4 and independent

then p(a b ) = 0.4 *0.4 = 0.16

There is very little difference between this answer and that computed in part (a). The difference is 0.16- 0.1596 =0.00004. Because this number is so small, it is quite reasonable to assume that A and B are independent for purposes of simplifying

For purposes of calculatingP(A?B), can we assume thatAandBof part (a) are independent to obtain essentially the correct probability

yes for purposes of caluclating p(A B) we can assme A and B of part (a) are independent

c) Now the total number of boards, N = 10. Without assuming independence, we may calculate the intersection of A and B in the same fashion as part (a). Note that the probability that A occurs is still equal to 0.4

P(A B) = P(B | A) ×P(A) =. 1/9(*0.4) =0.0444

so answer is no

The difference between the solutions in parts (c) and (b) is 0.16-0.0444=0.1156

What is the critical difference between the situation here and that of part (a)?

The critical difference is that the population size in part (a) is small compared to the random sample of two boards

When do you think that an independence assumption would be valid in obtaining an approximately correct answer toP(A?B)?

we see that the validity of the assumption of independence improves as the population size increases relative to the sample size.

so option a This assumption would be valid when the population is much larger than the sample size.

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