<Dd> P (A B) = P (B A) P (A) / P (B) = (0.01) (0.50) / (0.024) = 5 / 24 . </Dd> <P> Given that the item is defective, the probability that it was made by the third machine is only 5 / 24 . Although machine 3 produces half of the total output, it produces a much smaller fraction of the defective items . Hence the knowledge that the item selected was defective enables us to replace the prior probability P (A) = 1 / 2 by the smaller posterior probability P (A B) = 5 / 24 . </P> <P> Once again, the answer can be reached without recourse to the formula by applying the conditions to any hypothetical number of cases . For example, if 100,000 items are produced by the factory, 20,000 will be produced by Machine A, 30,000 by Machine B, and 50,000 by Machine C. Machine A will produce 1000 defective items, Machine B 900, and Machine C 500 . Of the total 2400 defective items, only 500, or 5 / 24 were produced by Machine C . </P> <P> The interpretation of Bayes' theorem depends on the interpretation of probability ascribed to the terms . The two main interpretations are described below . </P>

When does p (a and b) c = p(a c)p(b c)