In a shipment of laptop computers, 4% have a faulty keyboard, 6% have a faulty s

Question

In a shipment of laptop computers, 4% have a faulty keyboard, 6% have a faulty screen, and 1% have a faulty keyboard and a faulty screen. (a) What is the probability that a randomly selected laptop has at least one of these defects? (b) Given that a laptop has a faulty keyboard, what is the probability that it also has a faulty screen? (c) Given that a laptop does not have a faulty keyboard, what is the probability that its screen is also not faulty? (d) A test is used to detect faulty screens. Screens that really are faulty fail the test 98% of the time. However, screens that are not faulty fail the test 10% of the time. Given that a screen has failed the test, what is the probability that it is actually faulty? (e) Identify one advantage and one disadvantage about the test described in (d). Should the company be comfortable using this test? Justify your answer.

Explanation / Answer

let event of faulty keyboard =A ; of faulty screen =B

a) probabilty of at least one of these defects =P(AUB) =P(A)+P(B)-P(AnB) =0.04+0.06-0.01=0.09

b)probability=P(B|A) =P(AnB)/P(A) =0.01/0.04=1/4=0.25

c) P(Bc|Ac) =P(AcnBc) /P(Ac) =(1-P(AUB))/(1-P(A)) =(1-0.09)/(1-0.04)=0.91/0.96=91/96

d)probabilty that screen failed the test P(C) =0.98*P(A)+0.1*(1-P(A)) =0.98*0.04+0.1*0.96=0.1352

therefore probability that it is actually faulty given screen failed the test =P(A|C) =0.98*0.04/0.1352=0.2899

d) advantage : the test will reduce inspection cost to the company and higher chances that a defect free piece will pass the test

disadvantage:the test has lower amount of specificity which have high chance to report failure for a good pice

the company should not be comfortable because of uncertainity of faulty status of a screen even after getting a failure in test.

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