Answered: A lumber company has just taken…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 28EQ

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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) Compute P(A), P(B), and P(A ∩ B) (a tree diagram might help). (Round your answer for P(A ∩ B) to five decimal places.)

P(A)	=
P(B)	=
P(A ∩ B)	=

Are A and B independent?

Yes, the two events are independent.No, the two events are not independent.

(b) With A and B independent and P(A) = P(B) = 0.3, what is P(A ∩ B)?

How much difference is there between this answer and P(A ∩ 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 ∩ B), can we assume that A and B of part (a) are independent to obtain essentially the correct probability?

YesNo

(c) Suppose the lot consists of ten boards, of which three are green. Does the assumption of independence now yield approximately the correct answer for P(A ∩ B)?

YesNo

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.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 smaller in part (a).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 ∩ B)?

This assumption would be valid when there are fewer green boards in the sample.This assumption would be valid when there are more green boards in the sample. 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.

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