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 E_{1} be the event of time consumed by machine A, E_{2} be the event of time consumed by machine B and E_{3} 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|E_{1}) = P (defective item produced by A)

And P(X|E_{2}) = P (defective item produced by B)

And P(X|E_{3}) = P (defective item produced by C)

Now the probability that item produced by machine A, being given that defective item is produced, is P(E_{1}|A).

By using bayes’ theorem, we have:

11