Question

In a bolt factory 3 machines A, B, And C manufacturer 25,33 and 40 percent the total bolts manufactured. Of their 5,4 and 2 percent are defects respectively. A bolt is drawn and is found to be defective. Find the probability it was manufactured by either A or C.

Answer 1

Let N be total number of bolts manufactured.  (perhaps per year)

Machine A produces 25% of N = 0.25 N
Machine B  produces : 0.33 N  bolts
Machine C produces  0.4 N  bolts
Perhaps the remaining 2% are produced by some other machine.

Defects from machine A :  0.25 N * 5/100 = 0.0125 N
defects from machine B  :  0.33 N * 4/100 = 0.0132 N
defects from machine C :  0.40 N * 2/100 =  0.008 N

Total number of defects = (0.0125+0.0132+0.008) N = 0.0337 N

probability that a bolt, produced in the bolt factory, is found to be defective
            =  0.0337 N / N = 0.0337 or  3.37 %

P ( bolt from A / given that it is defective ) = 0.0125 N / 0.0337 N
              = 0.3709
P (bolt is from B / given that it is defective) = 0.0132 N / 0.0337 N
             = 0.3917
P (bolt is from C/ given that it is defective) = 0.008 N / 0.0337 N
            =  0.2374

Answer =  0.3709 + 0.2374  or  1 - 0.3917