Question

A construction company employs three sales engineers. Engineers 1,2, and 3 estimated the costs of 20%, 30% and 50%, respectively, of all jobs bid on by the company. For i=1,2,3, define Ei to be the event that a job is estimated by engineer i. The following probabilities describe the rates at which engineers make serious errors in estimating costs: P(error|E1)=0.03 P(error|E2)=0.04 P(error|E3)=0.01 (Give answers to three decimal places.)

a) If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer 1?

b) If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer 2?

c) If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer 3?

Answer 1

Date Page Given 1 Ezila gob is estimated by engineen 13 Ez : { À job is estimated by engineer 23 Ezu Ş A job is estimated by engineer 33 PLE,J = 0.2 PLEJ = 0.3 PLE) = 0.5 =0.5 = 0 - 2 -0.3 we also have, plexor Es 0.03. plerorl E 2) = o.o4 Pleerorl E3) = 0.01 Whore event Lerror) means that engineer serious ewor_in estimating costs. make Soun- kle will, Bayes's rules to find their probability: according to baye's rules PLBila)= pl Bina READ -

Date Page pl Bi) PLAIBI) ALBO PLATB.) *PlBx) platby) + + PlBkJ PLATEK) Baye's rule say that if we have kevent which are utually execulsive and exhaustive event mi such that pl Bi) + P(B2) t- + P(B) = 1 for, Even ELE2, E3 » PLEI) +P(E2) +P(E) >> 0.3+0.2+0.5-1 Now we use baye's rule we have :- PEIPS) - PLEJ) plerore PLE.) Plexror! Ed) +PLE2) Plexror|E2) + PLElle leveren. E3) 0.2X0:03 (0.246.03) +10.380.04) +10.5X6.01) 0.006 0:00 6 + 0.012 +0.005 >> 0.006 0.023 6/23 = 0.26

Probability that the ever was made by E 1=0.26= ba Plezl ever) PLE2) Pl error E2 PLEJ Plerror E1) + PLE2) Pleror / ET +PLE3) Plevorleg) 0.3 x 0.04 0.023 12 0.52 23 Probability that the exercer was made by E2=0.52 ON Plezlervor) PLE3) Plerror/ E3) PLE) Plevrozles) + PLE2) PLEJOX|E2) tples pleoort €3) O.Sxo.ol 2023 5 23 25 = 0.217 m 0.22 probability that the encorat was made by E3= 0.22