Question

A factory has two machines, A and B. Past records show that the machine A
produces 30% of the total output and the machine B, the remaining 70%.
Machine A produces 5% defective articles and Machine B produces 1%
defective items. An item is drawn at random and found to be defective.
What is the probability that it was produced (i) by the machine A, and
(ii) by the Machine B?
Ans.: (1) 0.682 (ii) 0.318)​

Answer 1

Answer:

Let E1E1 be the event that the item is produced by machine A. P(E1E1) = 60% = 60100=3560100=35

Let E2E2 be the event that the item is produced by machine A. P(E2E2) = 40% = 40100=2540100=25

Let A be the even that we choose a defective item at random:

P (the defective item came from machine A) = P (A|E1E1) = 2% = 21002100

P (the defective item came from machine B) = P (A|E2E2) = 1% = 11001100

We need to find the probability that a defective item randomly selected was produced by machine B

We can use Baye's theorem, according to which P(E2|A)=P(E2)(P(A|E2)P(E1)P(A|E1)+P(E2)P(A|E2)P(E2|A)=P(E2)(P(A|E2)P(E1)P(A|E1)+P(E2)P(A|E2)

P (E2E2|A) = 1100.351100.35+2100.251100.351100.35+2100.25 = 25006500+250025006500+2500=28=14