02 Historical data at Acme Manufacturing estimate 5% chance that a batch of widgets will be unacceptable (bad). A bad batch has 15% defective items, and a good batch includes only 4% defective items. Letting α = 0₁ and a = 0₂ represent a good and a bad batch, respectively, the associated prior probabilities are given as P{a = 0₁} = 0.95 and P{a = 0₂} = 0.05. Instead of shipping batches based solely on prior probabilities, a test sample of two items is used, giving rise to three possible outcomes: (1) both items are good (Z1), (2) one item is good (Z2), and (3) both items are defective(Z3). a) Determine the posterior probabilities P{0i|z;}, i = 1,2,3. b) Suppose the manufacturer ships batches to two customers A and B. The contracts specify that the defectives for A and B should not exceed 5% and 8%, respectively. A penalty of $100 is incurred per percentage point above the maximum limit. Supplying better-quality batches than specified by the contract costs the manufacturer $50 per percentage point. Develop the associated decision tree and determine a priority strategy for shipping the batches.