A certain firm has plants A, B, and C producing respectively 35%, 15%, and 50% of the total output. The probabilities of a non-defective product are, respectively, 0.75, 0.95, and 0.85. A customer receives a defective product. What is the probability that it came from plant C?

Answer:There is a 44.12% probability that the defective product came from C.Step-by-step explanation:This can be formulated as the following problem:What is the probability of B happening, knowing that A has happened.It can be calculated by the following formula[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.-In your problem, we have:P(A) is the probability of the customer receiving a defective product. For this probability, we have:[tex]P(A) = P_{1} + P_{2} + P_{3}[/tex]In which [tex]P_{1}[/tex] is the probability that the defective product was chosen from plant A(we have to consider the probability of plant A being chosen). So:[tex]P_{1} = 0.35*0.25 = 0.0875[/tex][tex]P_{2}[/tex] is the probability that the defective product was chosen from plant B(we have to consider the probability of plant B being chosen). So:[tex]P_{2} = 0.15*0.05 = 0.0075[/tex][tex]P_{3}[/tex] is the probability that the defective product was chosen from plant B(we have to consider the probability of plant B being chosen). So:[tex]P_{3} = 0.50*0.15 = 0.075[/tex]So[tex]P(A) = 0.0875 + 0.0075 + 0.075 = 0.17[/tex]P(B) is the probability the product chosen being C, that is 50% = 0.5.P(A/B) is the probability of the product being defective, knowing that the plant chosen was C. So P(A/B) = 0.15.So, the probability that the defective piece came from C is:[tex]P = \frac{0.5*0.15}{0.17} = 0.4412[/tex]There is a 44.12% probability that the defective product came from C.