A quality-control program at a plastic bottle production line involves inspectin

Question

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 0.98 that it will fail the inspection. If a bottle does not have a flaw, the probability is 0.995 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 bottle 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.

Explanation / Answer

A) P(having flow | fails inspection) = P(having flow and failing inspection) / P(failing inspection)

= 0.0002x0.98/(0.0002x0.98 + 0.9998x0.005)

= 0.0377

B) i) Most bottles that fail inspection do not have a flaw

C) P(doesnot have flaw | passes inspection) = P(passes inspection and does not have a flaw)/ P(passes inspection)

= 0.9998x0.995/(0.9998x0.995 + 0.0002x0.02)

= 0.999996

D) ii) Most bottles that pass inspection do not have a flaw

E) We donot want any defective bottles passing the test even if there are some non defective bottles that doesnt pass the test. This makes sure that the quality of bottles that pass the test ave very likely to be good even if we have some minor losses in terms of some flawless bottles being rejected.

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