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.
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.
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