a. A certain company sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event "late delivery" by L) . If a record of an overnight mailing is randomly selected from the company's file, what is the probability that the parcel went via E1 and was late? b. Suppose that 50% of the overnight parcels are sent via express mail service E2 and the remaining 10% are sent via E3. Of those sent via E2, only 1% arrive late, whereas 5% of the parcels handled by F3 arrive late. What is the probability that a randomly selected parcel arrived late? c. If a randomly selected parcel has arrived on time, what is the probability that it was not sent via E1?

Answer : Step 1 : Given A certain company sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time. a). A certain company sends 40% of its overnight mail parcels via express mail service E1.Of these parcels, 2% arrive after the guaranteed delivery time Here 40% = 40/100 = 0.4 and 2% = 2/100 = 0.02 So P(certain company sends via E1 ) = 0.4 Then P(arrive late) = 0.02 Both events are independent. P(certain company sends via E1 U arrive late ) Let the probability that the parcel went via E1 and was late. = P(certain company sends via E1) P(arrive late) = (0.4) (0.02) = 0.008 Therefore the probability that the parcel went via E1 and was late is 0.008. Step 2 : b). Suppose that 50% of the overnight parcels are sent via express mail service E2 and the remaining 10% are sent via E3. Of those sent via E2, only 1% arrive late, whereas 5% of the parcels handled by F3 arrive late. E2 = 50% of the overnight parcels = 50/100 =0.5 E3 = 10% = 10/100 = 0.1 P(certain company sends via E2 ) = 0.5 Then P(arrive late) = 0.1 P(certain company sends via E2 U arrive late ) = P(certain company sends via E2) P(arrive late) = (0.5) (0.1) = 0.05 Then we have to find the probability that a randomly selected parcel arrived late. So P(E2* E3 * E4) =P(E2* E3 * E4) =P(0.5 * 0.1 * 0.05) =0.0025 Therefore the probability that a randomly selected parcel arrived late is 0.0025