statistics

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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 (use
Bayes’s Rule).
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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