Problem 2-193 Four technicians (A, B, C and D) repair broken cars but sometimes

Question

Problem 2-193 Four technicians (A, B, C and D) repair broken cars but sometimes they make incomplete repair. Suppose technician A repairs 20% of the broken cars and makes incomplete repair ! time in 20; technician B repairs 60% of broken cars and makes incomplete repair 1 time in 20; technician C repairs 15% of broken cars and makes incomplete repair l in 10 times; technician D repairs 5% of broken cars and makes incormplete repair 1 in 20 times. If a given broken car was repaired and upon final inspection it was found that the repair was incomplete, what the probability that this car was repaired by technician A?

Explanation / Answer

By theorem of total probability,

P(incomplete) = P(incomplete | A)*P(A) + (incomplete | B)*P(B) + (incomplete | C)*P(C) + (incomplete | D)*P(D)

= 0.20(1/20) + 0.6(1/20) + 0.15(1/10) + 0.05(1/20)

= 0.0575

By Bayes' theorem,

P(A | incomplete) =  P(incomplete | A)*P(A) / P(incomplete)

= 0.20(1/20) / 0.0575

= 0.1739

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