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 0.991 that it will fail the inspection. If a bottle does not have a flaw, the probability is 0.98 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)?
1) Most bottles that fail inspection do not have a flaw
2) 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)?
1) Most bottle that fail inspection do have a flaw
2) 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.