A certain company sends 50% of its overnight mail parcels via express mail servi

Question

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

Explanation / Answer

a)

P(went via E1 and late)=P(went via E1)*P(late|via E1) =0.5*0.01 =0.005

b)

P(parel arrived late)=P(went via E1 and late)+P(went via E2 and late)+P(went via E3 and late)

=0.5*0.01+0.1*0.02+0.4*0.05 =0.027

c)

P(arrival on time) =1-P(arrived late) =1-0.027 =0.973

hence P(not sent from E1|arrived on time) =1-P(sent from E1|arrived in time)

=1-P(sent from E1 and on time)/P(arrived on time)=1-(0.5*(1-0.01))/0.973=0.491264

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