Question

.A certain federal agency employs three consulting firms (A, B and C) with probabilities 0.4, 0.35, 0.25, respectively. From past experience, it is known that the probabilities of cost overrun for the firms are 0.05, 0.03, and 0.15 respectively. Suppose a cost overrun is experienced by the agency. a) What is the probability that the consulting firm involved is company C? b) What is the probability that it is company A?

Answer 1

Solution:

It is given:

P(A)=0.4

P(B)=0.35

P(C)=0.25

Let L denotes the event that the company experience cost overrun.

It is given:

P(L | A)=0.05

P(L | B)=0.03

P(L | C)=0.15

a) We find the probability that the consulting firm involved is company C, i.e. we need to find $P(C \mid L)$ .

Using Bayes' theorem, we get the required probability :

$\begin{aligned} P(C \mid L) &=\frac{P(L \mid C) P(C)}{P(L \mid C) P(C)+P(L \mid B) P(B)+P(L \mid A) P(A)} \\ &=\frac{0.15 \cdot 0.25}{0.15 \cdot 0.25+0.03 \cdot 0.35+0.05 \cdot 0.4} \\ &=\frac{0.0375}{0.068} \\ & \approx 0.551 \end{aligned}$

b) We find the probability that the consulting firm involved is company A, i.e. we need to find $P(A \mid L)$ .

Using Bayes' theorem, we get the required probability :

$\begin{aligned} P(A \mid L) &=\frac{P(L \mid A) P(A)}{P(L \mid C) P(C)+P(L \mid B) P(B)+P(L \mid A) P(A)} \\ &=\frac{0.05 \cdot 0.4}{0.15 \cdot 0.25+0.03 \cdot 0.35+0.05 \cdot 0.4} \\ &=\frac{0.02}{0.068} \\ & \approx 0.294 \end{aligned}$