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.
Let Fl denote the event that a bottle has a flaw. Let F denote the event that a bottle fails inspection. We are given P(Fl) = 0.0002, P(F|Fl) = 0.995, and 
= 0.99.
(b) i. Given that a bottle failed inspection, the probability that it had a flaw is only 0.01952.
(c)
(d) ii. Given that a bottle passes inspection, the probability that is has no flaw is 0.999999.
(e) The small probability in part (a) indicates that some good bottles will be scrapped. This is not so serious. The important thing is that of the bottles that pass inspection, very few should have flaws. The large probability in part (c) indicates that this is the case.
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