A lumber company has just taken delivery on a shipment of 10,000 2 4 boards. Sup

Question

A lumber company has just taken delivery on a shipment of 10,000 2 4 boards. Suppose that 30% of these boards (3000) 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}.

A lumber company has just taken delivery an a shipment of 10 000 2 x 4 boards Suppose that 30% of these boards 3000 are actually too green to be used in first-quality construction. Two boards are se ected at random one after the other. Let A-(the first board is green) and B(the second board is green) (a) Compute PiA), P), and PAn B) (a tree diagram might help). Round your answer for PIAn) to five decimal places.) P[A) (8) = Are A and 5 independent? O Yes, the two events are independent O No, the two evants are not independent (b) With A and B independent and PIA)P(B)0.2, what is PAn B)? How much difference is there between this answer and PAn B) in part (a)? There Is no differance. There is very little difference. There is a very large difference. For purposes of calculating PA n B), can we assume that A and B of part (a)are independent to obtain essentially the correct probability? O Yes O No (c) Suppose the lot consists or ten boards, of which three are green. Does the assumption of independence now yield approximately the correct answer for PIA n B)? Yes O No What is the critical ditference between the situation here and that of part (a)? O The critical difference is that the percentage of green boards is smaller in part (a) O 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 population slze In part (a) Is small 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(AnB) This assumption would be valid when the sample size is very large. This assumption would be valid when the population is much larger than the sample size. This assumption would he valid when there are more greaen hoards in the sampla. This assumption wauld he valid when there are fawer green boards in the sample.

Explanation / Answer

a) P(A) =3000/10000 =0.30

P(B) =P(firsrt board green and second green +first board not green and second is )

=(3000/10000)*(2999/9999)+(7000/10000)*(3000/10000)=0.3

P(A n B) =P(firsrt board green and second green) =(3000/10000)*(2999/9999) =0.08998

No the two events are not independent.

b) P(A n B) =PA)*P(B) =0.3*0.3 =0.09

there is a very little differnce

Yes

c) No

the critical difference is population size in part (a) is huge compared to random sample of two bards

when the population size is much larger than the sample size.

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