Suppose that you\'re thinking about buying a used R-car at Honest Abe\'s. In ord

Question

Suppose that you're thinking about buying a used R-car at Honest Abe's. In order to make an informed decision you look up the records in an auto magazine and find that 30% of these cars have a faulty transmission. To get more information you hire a mechanic who is excellent: Of all the faulty cars he has examined in the past he correctly judged that 90% were "faulty" and only erroneously judged 10% as "OK." He's almost as good at judging good cars: Of all the good cars he's correctly judged that 80% were "good" and only erroneously judged 20% as "faulty."

What is the probability that the R-car you're thinking of buying has a faulty transmission if the mechanic judges it to be "faulty"?

.72

.80

.16

.66

.30

.72

.80

.16

.66

.30

Explanation / Answer

From given information, P(Faulty)=0.30, P(OK)=0.70, P(judges faulty|faulty)=0.90, P(judges OK|OK)=0.80

Use, Bayes theorem to compute the following probability.

P(Faulty|judges faulty)=P(Faulty)P(judges faulty|faulty)/{P(Faulty)P(judges faulty|faulty)+P(OK)P(judges OK|OK)}

=(0.30*0.90)/(0.30*0.90+0.70*0.80)

=0.3253

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