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3. (40 pts) A dot. B, and C). Assume that Warehouses A, B, and C ship 30%, 20% and 50% com company ships products from three different Warehouses (A, of the dot.com\'s sales, respectively Based on customer complaints, it appears that 3% of the shipments com- ing from A are somewhat faulty, as are 5% of the shipments from B, and 2% coming from C. Suppose a customer is mailed an order and calls in a complaint the next day. a) What is the probability that a customer complains about a faulty item? b) What is the probability the item came from Warehouse C?

Solution

3. let A be the event that the order was shipped from warehouse A.

let B be the event that the order was shipped from warehouse B.

let C be the event that the order was shipped from warehouse C.

let F be the event that the order is faulty.

so F|A is the event of faulty order from A, F|B is the event of faulty order from B, F|C is the event of faulty order from C

by question it is known that

P[A]=0.3 P[B]=0.2   P[C]=0.5

P[F|A]=0.03 P[F|B]=0.05 P[F|C]=0.02

a) the probability that the customer complains about a faulty item is

P[F]=P[A]*P[F|A]+P[B]*P[F|B]+P[C]*P[F|C]=0.3*0.03+0.2*0.05+0.5*0.02=0.029   [answer]

b) probability that the faulty item came from warehouse C is

P[C|F]=P[C]*P[F|C]/P[F]    [by Bayes\' theorem]

         =0.5*0.02/0.029=0.3448 [answer]