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 0.00025. If a bottle has a flaw, the probability is 0.99 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

Here we are given that 0.00025 of the bottles have got flaw in them. Therefore,

P( Flaw) = 0.00025, Therefore P( No Flaw) = 1 - P(Flaw) = 1 - 0.00025 = 0.99975

The probability that bottle fails the inspection given that it has a flaw is 0.99. Therefore,

P( Fail | Flaw) = 0.99. Therefore P( Pass | Flaw) = 0.01

Probability that the bottle pass the inspection given that the bottle does not have a flaw is given as:

P( Pass | No flaw ) = 0.995, Therefore P( Fail | No Flaw) = 0.005

a) Here we have to find the probability that the bottle has a flaw given that the bottle fails the inspection

P( Fail ) = P( Fail | No Flaw) P( No flaw) + P( Fail | Flaw) P( Flaw)

P( Fail) = 0.005*0.99975 + 0.99*0.00025 = 0.00499875 + 0.0002475 = 0.00524625

Now Using Bayes theorem we have:

P( Fail | Flaw)P(Flaw) = P( Flaw | Fail )P(Fail)

Putting all the values we get:

0.99*0.00025 = P( Flaw | Fail )*0.00524625

Therefore,

P( Flaw | Fail ) = 0.99*0.00025 / 0.00524625 = 0.0472

Therefore the required probability here is 0.0472

b) Clearly from the computed probability we see that about 5 in every 100 bottles that fail the test would have a flaw. Therefore i) most bottles that fail the inspection do not have a flaw is the correct answer

c) P( Pass) = 1 - P(Fail) = 1 - 0.00524625 = 0.99475375

Now using bayes theorem we get:

P( Pass | No flaw)P(No flaw) = P( No flaw | Pass )P(Pass)

Putting all the values we get:

0.995*0.99975 = P( No flaw | Pass )*0.99475375

Therefore, we get:

P( No flaw | Pass ) = 0.995*0.99975/ 0.99475375 = 0.999997

d) Here as the probability computed is too high, therefore most bottles that pass the inspection do not have a flaw.

Therefore (ii) is correct here

e) Small probability in part a) is not a problem as long as the probability in part (c) is large enough because the total probability of plastic bottle being defective is very very low ( 0.00025 ) and the test is able to bring it till 0.047 detection which is much better also it is important that we catch the bad bottles and therefore it is important to minimize the probability of having a bottle with flaw given it passes the inspection which is not the case here, that probability is way too large.

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