Historical data at Acme Manufacturing estimate a 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 a = θ₁ and a = θ₂ represent a good and a bad batch, respectively, the associated prior probabilities are given as

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{θ_i|z_j }, i = 1, 2; j = 1, 2, 3.

(b) Suppose that 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.

Transcribed Image Text:

P{a = 0₁} = 95 and P { a = 0₂ } = .05