Question

In a screw factory machines A, B, C manufacture respectively 30%, 40% and 30% of the total output of these 2%, 4% and 6% percent are defective screws. A screw is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by Machine C?

Answer 1

Let E₁, E₂, E₃ and A be the events defined as follows:

E₁ = Screw is manufactured by machine A

P`("E"_1) = 30/100`

E₂ = Screw is manufactured by machine B

P`("E"_2) = 40/100`

E₃ = Screw is manufactured by machine C

P`("E"_3) = 30/100`

A = Screw is defective

P`("A"/"E"_1)` = Probability that the defective screw is manufactured by machine A

= `2/100`

Similarly

P`("A"/"E"_2) = 4/100` and P`("A"/"E"_3) = 6/100`

Required probability = Probability that the bolt is manufactured by machine C given that the bolt drawn is defective = P`("E"_3/"A")`

∴ `"P"("E"_3/"A") = ("P"("E"_3) xx "P"("A"/"E"_3))/("P"("E"_1) xx "P"("A"/"E"_1) + "P"("E"_2) xx "P"("A"/"E"_2) + "P"("E"_3) xx "P"("A"/"E"_3))`

= `(30/100 xx 6/100)/(30/100 xx 2/100 + 4/100 xx 4/100 + 3/100 xx 6/100)`

= `(180/(10,000))/(60/(10,000) + 160/(10,000) + 180/(10,000))`

= `180/400`

P`("E"_3/"A")` = 0.45