Question

the following information pertains to three shipping terminals operated by a certain corporation. their internal auditor randomly selects one set of shipping documents, ascertaining that the set selected contains an error. find the probability that the error occurred in the air terminal.
which of the following gives the probability that the error occurred in the air terminal?
a. 0.26
b. 0.58
c. 0.22
d. 0.45
terminal|percentage of cargo handled|percent error
land|50|5
air|40|9
sea|10|17

Explanation:

Step1: Calculate the overall probability of an error

Let $P(L)$ be the probability of cargo being handled at the Land terminal, $P(A)$ be the probability of cargo being handled at the Air terminal, $P(S)$ be the probability of cargo being handled at the Sea terminal. Let $P(E|L)$ be the probability of an error given it is at the Land terminal, $P(E|A)$ be the probability of an error given it is at the Air terminal, $P(E|S)$ be the probability of an error given it is at the Sea terminal.
$P(L)=0.5$, $P(A) = 0.4$, $P(S)=0.1$, $P(E|L)=0.05$, $P(E|A)=0.09$, $P(E|S)=0.17$.
By the law of total - probability, $P(E)=P(L)P(E|L)+P(A)P(E|A)+P(S)P(E|S)$.
$P(E)=(0.5\times0.05)+(0.4\times0.09)+(0.1\times0.17)$
$P(E)=0.025 + 0.036+0.017$
$P(E)=0.078$.

Step2: Calculate the probability that the error occurred in the Air terminal given an error

By Bayes' theorem, $P(A|E)=\frac{P(A)P(E|A)}{P(E)}$.
We know $P(A)P(E|A)=0.4\times0.09 = 0.036$ and $P(E)=0.078$.
$P(A|E)=\frac{0.036}{0.078}\approx0.46$.

Answer:

D. 0.46