A factory has three machines X,Y and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defectives bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?
Given:
Factory X has produced 1000 bolts per day
Factory Y has produced 2000 bolts per day
Factory Z has produced 3000 bolts per day.
Let us assume U1, U2, U3 and A be the events as follows:
U1 = choosing factory X
U2 = choosing factory Y
U3 = choosing factory Z
A = choosing a defective bolt
⇒ 
⇒ 
⇒ 
From the problem:
⇒ P(A|U1) = P(Choosing defective bolt from factory X)
⇒ 
⇒ P(A|U2) = P(Choosing defective bolt from factory Y)
⇒ 
⇒ P(A|U3) = P(Choosing defective bolt from factory Z)
⇒ 
Now we find
P(U1|A) = P(The chosen bolt is from factory X)
Using Baye’s theorem:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
∴ The required probability is 
.