An item is manufa

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