7. Historical data at Acme Manufacturing estimate a 5% chance that a manufactured batch of widgets will be unacceptable (bad). A bad batch has 15% defective items, and a good batch includes only 4% defective items. Letting a = lh (= 92) indicate that the batch is good (bad), the associated prior probabilities are given as P{a = 8t } = .95 and P{a = 82} = .05 Instead of shipping batches based solely on prior probabilities, a test sample of two items is used, which gives rise to three possible outcomes: (1) both items are good (Zl)' (2) one item is good (Z2), and (3) both items are defective (Z3)'

(a) Determine the posterior probabilities P{9i lzJ, i = 1,2; j = 1,2,3.