Question

In a bolt factory, three machines A, B, and C manufacture 25%, 35% and 40% of the total production respectively. Of their respective outputs, 5%, 4% and 2% are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine C.

Answer 1

	Machine A	Machine B	Machine C
% of total production	25%	35%	40%
% defective of their output	5%	4%	2%

∴ P(A) = `(25)/(100) = (1)/(4)`,

P(B) = `(35)/(100) = (7)/(20)`

and P(C) = `(40)/(100) = (2)/(5)`

P(D/A) = `(5)/(100) xx (1)/(4) = (1)/(80)`

P(D/B) = `(4)/(100) xx (7)/(20) = (7)/(500)`

P(D/C) = `(2)/(100) xx (2)/(5) = (1)/(125)`

P(C/D) = `("P"("C") ·"P"("D"/"C"))/( ("P"("A") ·"P("D"/"A") + "P"("B") ·"P"("D"/"B) +"P"("C") ·"P"("D"/"C"))`

= `((2)/(5) xx (1)/(125))/ ((1)/(4) xx (1)/(80) + (7)/(20) xx (7)/(500) + (2)/(5) xx (1)/(125))`

=  `((2)/(625))/((250 + 392 + 256)/(80000))`

= `(2xx 80000)/(625 xx 898) = (128)/(449) = 0.2851`