A certain company sends 50% of its overnight mail parcels via express mail service E1. 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.
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?
B. 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 is was not sent via E1?

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?

50% are chosen via E1, and of those, 1% were late. So

[tex]p = 0.5*0.01 = 0.005[/tex]

0.005 = 0.5% probability that the parcel went via El and was late

B. What is the probability that a randomly selected parcel arrived late?

1% of 50%(via E1)

2% of 10%(via E2)

5% of 40%(via E3).

So

[tex]p = 0.01*0.5 + 0.02*0.1 + 0.05*0.4 = 0.027[/tex]

0.027 = 2.7% probability that a randomly selected parcel arrived late.

C. If a randomly selected parcel has arrived on time, what is the probability that is was not sent via E1?

Conditional Probability.

Event A: Late

Event B: Not sent via E1

0.027 = 2.7% probability that a randomly selected parcel arrived late, which means that [tex]P(A) = 0.027[/tex]

Late and not sent via E1:

2% of 10%(via E2)

5% of 40%(via E3).

So

[tex]P(A \cap B) = 0.02*0.1 + 0.05*0.4 = 0.022[/tex]

Desired probability:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.022}{0.027} = 0.8148[/tex]

0.8148 = 81.48% probability that is was not sent via E1