In a factory, machine A produces 30% of the total output, machine B produces 25% and the machine C produces the remaining output. If defective items produced by machines A, B, C are 1%,1.2%, 2% respectively. Three machines working together produce 10000 items in a day. An item is drawn at random form a day’s output and found to be defective. Find the probability that it was produced by machine B?
Let us assume U1, U2, U3 and A be the events as follows:
U1 = choosing Machine A
U2 = choosing Machine B
U3 = choosing Machine C
A = Producing a defective output
From the problem:
⇒ 
⇒ 
⇒ 
⇒ P(A|U1) = P(Producing defective output from Machine A)
⇒ 
⇒ P(A|U2) = P(Producing defective output from Machine B)
⇒ 
⇒ P(A|U3) = P(Producing defective output from Machine C)
⇒ 
Now we find
P(U2|A) = P(The found defective item is produced by Machine B)
Using Baye’s theorem:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
∴ The required probability is 
.