In a factory machines A, B, C manufacture 25%, 35% and 40% of the total their output and 6%, 3%, and 2% are defective. A bolt is drawn at random. If the selected bolt found to be defective, what is the probability that it is manufactured by machine C
Let's find the probability that the bolt is defective and then is manufactured by machine C.
Let:
P(A) = Probability that bolt is manufactured by machine A = 25%
P(B) = Probability that bolt is manufactured by machine B = 35%
P(C) = Probability that bolt is manufactured by machine C = 40%
P(D|A) = Probability that bolt is defective given it is manufactured by machine A = 6%
P(D|B) = Probability that bolt is defective given it is manufactured by machine B = 3%
P(D|C) = Probability that bolt is defective given it is manufactured by machine C = 2%
We want to find P(C|D), the probability that the bolt is from machine C given that it is defective.
According to Bayes' theorem:
P(C|D) = P(C and D) / P(D)
P(C and D) = P(D|C) * P(C)
P(D) = P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)
Plugging the values:
P(C and D) = 0.02 * 0.40
P(D) = 0.06 * 0.25 + 0.03 * 0.35 + 0.02 * 0.40
Now, we can find P(C|D):
P(C|D) = (0.02 * 0.40) / (0.06 * 0.25 + 0.03 * 0.35 + 0.02 * 0.40) = 0.008 / 0.0225 = 0.3556
Therefore, the probability that the selected defective bolt is manufactured by machine C is 0.3556 or 35.56%.