# Answer in Statistics and Probability for Shema Derrick #205149

.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?

**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}nnP(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)} \nn&=frac{0.15 cdot 0.25}{0.15 cdot 0.25+0.03 cdot 0.35+0.05 cdot 0.4} \nn&=frac{0.0375}{0.068} \nn& approx 0.551nnend{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}nnP(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)} \nn&=frac{0.05 cdot 0.4}{0.15 cdot 0.25+0.03 cdot 0.35+0.05 cdot 0.4} \nn&=frac{0.02}{0.068} \nn& approx 0.294nnend{aligned}”