An item is manufactured by three machines A,B and C. Out of the total number of items manufactured during a specified period, 50% are manufacture on machine A, 30% on B and 20% on C. 2% of the items produced on A and 2% of items produced on B are defective and 3% of these produced on C are defective. All the items stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?
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 = manufacturing defective item
⇒ 
⇒ 
⇒ 
⇒ P(A|U1) = P(Manufacturing defective item from machine A)
⇒ 
⇒ P(A|U2) = P(Manufacturing defective item from machine B)
⇒ 
⇒ P(A|U3) = P(Manufacturing defective item from machine C)
⇒ 
Now we find
P(U1|A) = P(The defective item is manufactured from machine A)
Using Baye’s theorem:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
∴ The required probability is 
.