Question

Q4) A desk lamp produced by The Luminar Company was found to be defective (D). There are three factories (A, B, C) where such desk lamps are manufactured. A Quality Control Manager is responsible for investigating the source of found defects. They know that factory A supplies 50% of the total lamps, factory B supplies 30% of the total lamps, and factory C supplies 20% of the total lamps. They also know that on average, 2.0% or factory A’s lamps are defective, 1.0% of factory B’s lamps are defective, and 3.0% of factory C’s lamps are defective. If a randomly selected lamp is defective, what is the probability that it was manufactured in factory A? If a randomly selected lamp is defective, what is the probability that it was manufactured in factory B? If a randomly selected lamp is defective, what is the probability that it was manufactured in factory C? If a randomly selected lamp is not defective, what is the probability that it was manufactured in factory A?

Answer 1

A: Event of a randomly selected desk lamp manufactured by Factory A

B : Event of a randomly selected desk lamp manufactured by Factory B

C: Event of a randomly selected desk lamp manufactured by Factory C

P(A) = 0.50 ; P(B) = 0.30 P(C) = 0.20

D: Event of a randomly selected desk lamp is defective

P(D|A) : Probability of a randomly selected desk lamp is defective given that the desk lamp manufactured by Factor A=0.02

P(D|B) = 0.01

P(D|C) = 0.03

If a randomly selected lamp is defective, probability that it was manufactured in factory A = P(A|D)

From Bayes theorem

P(A)P(D|A) = 0.50 x 0.02 = 0.01

P(B)P(D|B) = 0.30 x 0.01 = 0.003

P(C)P(D|C) = 0.20 x 0.03 = 0.006

P(D) = P(A)P(D|A)+P(B)P(D|B)+P(C)P(D|C) = 0.01+0.003+0.006=0.019

If a randomly selected lamp is defective, probability that it was manufactured in factory A = 0.526315789

If a randomly selected lamp is defective, probability that it was manufactured in factory B = P(B|D)

From Bayes theorem

If a randomly selected lamp is defective, probability that it was manufactured in factory B =0.157894737

If a randomly selected lamp is defective, probability that it was manufactured in factory C = P(C|D)

From Bayes theorem

If a randomly selected lamp is defective, probability that it was manufactured in factory C = 0.315789474

If a randomly selected lamp is not defective, probability that it was manufactured in factory A

P(D|A) =0.02 ; P(|A) =1-P(D|A)=1-0.02=0.98

P(D|B) = 0.01; P(|B) =1-P(D|B)=1-0.01=0.99

P(D|C) = 0.03 ; P(​​​​​​​|C) =1-P(D|C)=1-0.03=0.97

P(A)P(|A) = 0.50 x 0.98 = 0.49

P(B)P(|B) = 0.30 x 0.99 = 0.297

P(C)P(|C) = 0.20 x 0.97 = 0.194

P() = P(A)P(|A)+P(B)P(|B)+P(C)P(|C) = 0.49+0.297+0.194=0.981; (P() = 1- P(D) = 0.019 = 0.981)

If a randomly selected lamp is not defective, probability that it was manufactured in factory A = 0.499490316