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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?
Solution
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
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