Question

An accounting firm noticed that of the companies it audits, 85% show no inventory shortages, 10% show small inventory shortages, and 5% show large inventory shortages. The firm has devised a new accounting test for which it believes the following probabilities hold: P(company will pass test | no shortage) = .90 P(company will pass test | small shortage) = .50 P(company will pass test | large shortage) = .20

a. If a company being audited fails this test, what is the probability of a large or small inventory shortage?

b. If a company being audited passes this test, what is the probability of no inventory shortage?

Answer 1

P(   no inventory   ) =    0.85
P(   small inventory   ) =    0.1
P(    large inventory   ) =    0.05

P(   pass test   | no inventory)=   0.9
P(   pass test   | small inventory)=   0.5
P(   pass test   | large inventory)=   0.2

a)

P(   fail test   | no inventory)= 1-0.9 = 0.1
P(   fail test   | small inventory)= 1-0.5 = 0.5
P(   fail test   | large inventory)= 1-0.2 = 0.8

P(fail test) = P(no inventory) * P(fail test| no inventory) + P(small inventory) *P(fail test| small inventory) + P( large inventory)*P(fail test| large inventory) =                                    0.85*0.1+0.1*0.5+0.05*0.8=               0.175

P(small inventory| fail test) = P(small inventory)*P(fail test| small inventory)/P(fail test)=               0.1*0.5/0.175=       0.2857
P( large inventory| fail test) = P( large inventory)*P(fail test| large inventory)/P(fail test)=               0.05*0.8/0.175=       0.2286

required probability = 0.2857 + 0.2286 = 0.5143 (answer)

b)

P(pass the test) = 1 - P(fail) = 1 - 0.175 = 0.825

P(no inventory| pass) = P(no inventory)*P(pass| no inventory)/P(pass)=               0.85*0.9/0.825=       0.9273 (answer)