A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Given: let E1 be the event of time consumed by machine A, E2 be the event of time consumed by machine B and E3 be the event of time consumed by machine C. Let X be the event of producing defective items.
Then 
And 
And 
As we a headed coin has head on both sides so it will shows head.
Also P(X|E1) = P (defective item produced by A) 
And P(X|E2) = P (defective item produced by B) 
And P(X|E3) = P (defective item produced by C) 
Now the probability that item produced by machine A, being given that defective item is produced, is P(E1|A).
By using bayes’ theorem, we have:
