A machine produces parts that are either good (90%), slightly defective (2%), or defective (8%). Assume that a one-year warranty is given for the parts that are shipped to customers. Suppose that a good part fails within the first year with probability 0.01, while a slightly defective part fails within the first year with probability 0.10, and a defective part fails within the first year with probability 0.50.
(a) Suppose that there is no inspection machine. What is the probability that a customer receives a part that fails within the first year and therefore is entitled to a warranty replacement?
(b) Now suppose produced parts get passed through an automatic inspection machine, which is able to detect any defective parts; these parts are then discarded. What is the probability that a customer receives a part that fails within the first year and therefore is entitled to a warranty replacement?
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