Answer :

Let events E_{1}, E_{2}, E_{3} be the following:

E_{1}: event that item is manufactured by machine A

E_{2}: event that item is manufactured by machine B

E_{3}: event that item is manufactured by machine C

Clearly, E_{1}, E_{2} and E_{3} 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|E_{1}) is the probability of the item drawn is defective given that it is manufactured on machine A = 2%

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

P(E|E_{3}) 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(E_{1}|E) is the probability that the item is drawn is defective and it was manufactured on machine A

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