A quality-control program at a plastic bottle production line involves inspecting finished bottles for flaws such as microscopic holes. The proportion of bottles that actually have such a flaw is only 0.0002. If a bottle has a flaw, the probability is 0.995 that it will fail the inspection. If a bottle does not have a flaw, the probability is 0.99 that it will pass the inspection.

a. If a bottle fails inspection, what is the probability that it has a flaw?

b. Which of the following is the more correct interpretation of the answer to part (a)?

i. Most bottles that fail inspection do not have a flaw.

ii. Most bottles that pass inspection do have a flaw.

c. If a bottle passes inspection, what is the probability that it does not have a flaw?

d. Which of the following is the more correct interpretation of the answer to part (c)?

i. Most bottles that fail inspection do have a flaw.

ii. Most bottles that pass inspection do not have a flaw.

e. Explain why a small probability in part (a) is not a problem, so long as the probability in part (c) is large.