The table below shows the defect rate for components manufactured at three facilities. Facility | Defect rate Facility…

To find z in terms of P, determine the proportion of defective components contributed by each facility.

Let N be the total number of components in the shipment. Since Facility C produces z% of the components, the remaining (100 - z)% is produced by Facility A and Facility B combined. Facility B produces twice as many components as Facility A, so if Facility A produces x% of the total shipment, Facility B produces 2x% of the total shipment.

Therefore:

x + 2x = 100 - z

3x = 100 - z

x = (100 - z) / 3

Thus, the percentage of the shipment produced by each facility is:

- Facility A: (100 - z) / 3 %

- Facility B: 2(100 - z) / 3 %

- Facility C: z%

Now calculate the number of defective components from each facility:

- Defective from A: N · [(100 - z) / 300] · 0.02 = N · [2(100 - z) / 30,000]

- Defective from B: N · [2(100 - z) / 300] · 0.03 = N · [6(100 - z) / 30,000]

- Defective from C: N · (z / 100) · 0.05 = N · (15z / 30,000)

The total number of defective components is:

[N / 30,000] · [2(100 - z) + 6(100 - z) + 15z] = [N / 30,000] · [8(100 - z) + 15z] = [N / 30,000] · (800 + 7z)

The conditional probability P that a randomly selected defective component came from Facility B is the ratio of defective components from Facility B to the total number of defective components:

P = [6(100 - z)] / (800 + 7z) = (600 - 6z) / (800 + 7z)

Now, solve this equation for z in terms of P:

P(800 + 7z) = 600 - 6z

800P + 7P · z = 600 - 6z

7P · z + 6z = 600 - 800P

z(7P + 6) = 600 - 800P

z = (600 - 800P) / (7P + 6)

Choice B is correct.

Choice A is incorrect because it represents the probability that the defective component was from Facility A.

Choice C is incorrect because it omits the factor of 2 in the numerator for Facility B's production volume.

Choice D is incorrect because it fails to divide the remaining (100 - z)% by 3 when establishing the production volumes of Facilities A and B.