43 In a manufacturing plant three machines A B and C produce

43. In a manufacturing plant, three machines, A, B, and C, produce 40%, 35%, and 25%, respectively, of the total production. The company\'s quality-control department has determined that 1% of the items produced by Machine A, 1.5% of the items produced by Machine B, and 2% of the items produced by Machine C are defective. If an item is selected at random and found to be defective, what is the probability that it was produced by Machine B?

Solution

Let E₁ E₂ E₃ denote the events that the item produced by machine A, machine B, machine C, respectively.

Let D denote the item is defective then we have

P(E₁ ) =

P(E₂ ) =

P(E₃ ) =

P() = 1% = 0.01

P() = 1.5% = 0.015

P() = 2% = 0.02

The probability that an item selected at random and if found to be defective is given by

P(D) = E₁ D)

           = E₁) P(D/E₁)

           = (0.40)(0.01) + (0.35)(0.015) +(0.25)(0.02)

P(D)    = 0.0143

By using baye’s rule the probability the defective is produced by machine B is

P(E₂/D) =

                 =

                    =

                    = 0.3671

Therefore the probability that it was produced by machine B is 0.3671