Question

A certain delivery service offers both express (E) and standard delivery (S). Seventy-five percent of parcels are sent by standard delivery, and 25% are sent by express. Of those sent standard, 80% arrive the next day (N), and of those sent express, 95% arrive next day. A record of a parcel delivery is chosen at random from the company’s files.

a. What is the probability that the parcel was shipped express and arrived the next day?
b. What is the probability that it arrived the next day?
c. Given that the package arrived the next day, what is the probability that it was sent by express?

Answer 1

1)$P(N\cap E)=P(E)\cdot P(N|E)=0.25\cdot 0.95=\frac{19}{80}$

2)$P(N)=P(N\cap (E\cup S))=P((N\cap E)\cup (N\cap S))$ . Since $(N\cap E)\cap (N\cap S)=N\cap (E\cap S)=N\cap\varnothing=\varnothing$ , we have

$P((N\cap E)\cup (N\cap S))=P(N\cap E)+P(N\cap S)=$

$P(E)P(N|E)+P(S)P(N|S)=$

$=0.25\cdot 0.95+0.75\cdot 0.8=\frac{67}{80}$

3)$P(E|N)=\frac{P(E\cap N)}{P(N)}=\frac{19}{67}$

Answer: 1)$\frac{19}{80}$ 2)$\frac{67}{80}$ 3)$\frac{19}{67}$