Question

A lumber company has just taken delivery on a shipment of 10,000 2 ✕ 4 boards. Suppose that 10% of these boards (1000) 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) = 0.1, 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 one 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 percentage of green boards is smaller in part (a).    The critical difference is that the percentage of green boards is larger in part (a).The critical difference is that the population size in part (a) is huge 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 to P(A ∩ B)?
This assumption would be valid when the population is much larger than the sample size.This assumption would be valid when the sample size is very large.    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.

Answer 1

a)

P(A) = 0.1

P(B) =

Let B be the event that the second board is green.

The probability of is obtained below:

The probability of B is 0.1.

(a.3)

Consider,

Part a.3

The probability of is 0.00999.

(b.1)

It is given that the events A and B are independent with the probabilities

By the multiplication rule for the two independent events,

Part b.1

There is very little difference.

From the part a, the value of is 0.00999.

But, the value of is 0.01 when.

Thus, there is very little difference in the values of.

Yes, it can be possible to assume that the events A and B of part (a) are independent to obtain essentially the correct probability.

c)

By the conditional probability of the two events A and B,

Use to compute the probability of

By the multiplication rule, the two events A, B are said to be independent if,

Thus, the value of probability obtained by using and differs.

Part c.1

No, the assumption of independence does not yield approximately the correct answer for

(c.2)

The values of probabilities of in the part a, by using the conditional probability is 0.00999 and then by using the multiplication rule for two independent events is 0.01.

Thus, the difference is .

The values of probabilities of in the part c, by using the conditional probability is 0 and then by using the multiplication rule for two independent events is also 0.01.

Thus, the difference is .

Part c.2

The critical difference is that the percentage of green boards is smaller in part (a).

By the assumption of the independence, the population size increases relatively to the sample size.

Part c.3

This assumption would be valid when the sample size is very large.

Part a.1

The probability of A is 0.1.

Part a.2

The probability of B is 0.1.

Part a.3

The probability of is 0.00999.

Part a.4

No, the two events A, B are not independent.

Part b.1

There is very little difference.

Part b.2

There is very little difference.

Part b.3

Yes, it can be possible to assume that the events A and B of part (a) are independent to obtain essentially the correct probability.

Part c.1

No, the assumption of independence does not yield approximately the correct answer for.

Part c.2

The critical difference is that the percentage of green boards is smaller in part (a).

Part c.3

This assumption would be valid when the sample size is very large.