A manufacturer has three machine operators and . The first operator produces defective items, whereas the other two operators and produce and defective items respectively. A is on the job for of the time, is on the job for of the time and is on the job for of the time. A defective item is produced, what is the probability that it was produced by A?
A manufacturer has three machine operators and . The first operator produces defective items, whereas the other two operators and produce and defective items respectively. A is on the job for of the time, is on the job for of the time and is on the job for of the time. A defective item is produced, what is the probability that it was produced by A?

Solution:

Let be the event of machine time consumption. be the event of machine time consumption Let and represent the event of machine using time. Let be the occurrence of defective items being produced.

Then

Because a headed coin has a head on both sides, it will display one.

Also defective item produced by

And defective item produced by

And defective item produced by

Now, given that a defective item is generated, the probability of an item manufactured by machine A is .

We have used Bayes’ theorem to arrive at our conclusion.

We may now retrieve the result by swapping the values.