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Question
In a bolt factory, machines X, Y and Z manufacture 20%, 35% and 45% respectively of the total output. Of their output 8%, 6% and 5% respectively are defective bolts. One bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured in machine Y?
Solution
Let E1, E2, E3 be the events of drawing of bolt produced by machine X, Y and Z respectively.
Let D be the event of drawing a defective bolt.
∴ P(E1) = `20/100`
P(E2) = `35/100`
P(E3) = `45/100`
and `"P"("D"/"E"_1) = 8/100`
`"P"("D"/"E"_2) = 6/100`
`"P"("D"/"E"_3) = 5/100`
By Bayes theorem
P(defective bolt is produced by machine Y)
`"P"("E"_2/"D") = ("P"("E"_2)"P"("D"/"E"_2))/("P"("E"_1)"P"("D"/"E"_1) + "P"("E"_2)"P"("D"/"E"_2) + "P"("E"_3)"P"("D"/"E"_3))`
= `(35/100 xx 6/100)/(20/100 xx 8/100 + 35/100 xx 6/100 + 45/100 xx 5/100)`
= `210/(160 + 210 + 225)`
= `210/595`
∴ `"P"("E"_2/"D") = 6/17`
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