A manufacturer has three machine operators A, B and C. The first operator A produces 1% of defective items, whereas the other two operators B and C produces 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job 30% of the time and C on the job for 20% of the time. All the items are put into one stockpile, and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by A ?

Let E₁ be the event that machine A is defective. P(E₁) = 1%

Let E₂ be the event that machine B is defective. P(E₂) = 5%

Let E₃ be the event that machine C is defective. P(E₃) = 7%

Let A be the event that the item chosen is defective.

P(the defective item is manufactured by A) = P(A|E₁) = 50%

P(the defective item is manufactured by B) = P(A|E₂) = 30%

P(the defective item is manufactured by C) = P(A|E₃) = 20%

Now, According to the Bayes’ theorem,

Given E₁, E₂, E₃…….E_n are mutually exclusive and exhaustive events, we can find the conditional probability P(E_i|A) for any event A associated with E_i is given by:

Therefore,

The probability that defective item is produced by A,

Hence, the probability that the defective item is produced by machine A is .