An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period, 50% are manufactured on A, 30% on B and 20% on C. 2% of the items produced on A and 2% of items produced on B are defective, and 3% of these produced on C are defective. All the items are stored at one go down. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

Let events E₁, E₂, E₃ be the following:

E₁: event that item is manufactured by machine A

E₂: event that item is manufactured by machine B

E₃: event that item is manufactured by machine C

Clearly, E₁, E₂ and E₃ are mutually exclusive and exhaustive events and hence, they represent a partition of sample space.

Given that:

Items manufactured on machine A = 50%

Items manufactured on machine B = 30%

Items manufactured on machine C = 20%

So,

Now, Let E be the event that ‘an item is defective’.

P(E|E₁) is the probability of the item drawn is defective given that it is manufactured on machine A = 2%

P(E|E₂) is the probability of the item drawn is defective given that it is manufactured on machine B = 2%

P(E|E₃) is the probability of the item drawn is defective given that it is manufactured on machine C = 3%

So,

Now, we have to find the probability that the item which is picked

up is defective, it was manufactured on machine A

We use Bayes’ theorem to find the probability of occurrence of an event A when event B has already occurred.

∴

P(E₁|E) is the probability that the item is drawn is defective and it was manufactured on machine A