I suppose the part about
"B1, B2, and B3, make 35%, 40%, and 25%"
should be taken to mean produces 35% of some product, produces 40%, and produces 25%. Then
Each machine has some probability of making a defective product - denote this event by . Then
We want to find the probability that a defective product was made by machine , i.e. . By definition of conditional probability and the law of total probability, we have
(def. of conditional probability)
(def. of conditional probability)
(law of total probability)
(def. of conditional probability; this result is also known as Bayes' theorem)
So we have