1. (a) Suppose 1 in 1000 persons has a certain disease. A test detects the disease in 95% of diseased persons. The test also "detects" the disease in 1% of healthy persons. With what probability does a positive test diagnose the disease?

(b) Machines M1, M2, M3 produce these proportions of a article: M1 : 5%, M2 : 25%, M3 : 70%. The probability the machines produce defective articles is M1 : 5%, M2 : 4%, M3 : 2%. What is the probability a random article was made by machine M1, given that it is defective?

(c) A machine M consists of three independent parts, M1, M2, and M3. Suppose that M1 functions properly with probability 9 10 , M2 functions properly with probability 9 10 , M3 functions properly with probability 8 10 , and that the machine M functions if and only if its three parts function. What is the probability for the machine M to malfunction?