A construction company employs three sales engineers. Engineers 1, 2, and 3 esti

Question

A construction company employs three sales engineers. Engineers 1, 2, and 3 estimate the costs of 30%, 20%, and 50%, respectively, of all of the jobs bid by the company. For i = 1, 2, 3 define Ai to be the event that a job is estimated by engineer I, and define E to be the event that a serious error is made in estimating the cost.

The following probabilities are known to describe the error rates of the engineers:

P(E|A1) = 0.01
P(E|A2) = 0.03
P(E|A3) = 0.02

If a particular bid results in a serious error in estimating the jobs cost:

A. What is the probability that the job was bid by engineer 1?

B. What is the probability that the job was bid by engineer 2?

C. What is the probability that the job was big by engineer 3?

Explanation / Answer

ans=

by baye's theorem,
P[Ai|E] = P[Ai].P[E|Ai] / sigma[P(Ai).P(E|Ai)]

so P[A1|E]
= [.3*.01] / [(.3*.01)+(.2*.03)+(.5*.02)]
= .003 / .019 = 3/19
P[A2|E] = .2*.03 / .019 = 6/19
P[A3|E] = .5*.02 / .019 = 10/19

B. DERIVATION FROM FUNDAS

suppose 1000 job costs hv been estimated
[1000 taken just to eliminate decimals ]

engr.1 likely to make 300*0.01 = 3 errors
engr.2 likely to make 200*0.03 = 6 errors
engr.3 likely to make 500*0.02 = 10 errors
total errors likely to be made = 19

thus if an error HAS been made,
most likely engineer to have made the error
is 3 , with a probability of 10/19

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