A certain federal agency employs three consulting ﬁrms (A, B and C) with probabilities 0.4, 0.35 and 0.25 respectively. From pas

Question

2 months ago

8

A certain federal agency employs three consulting ﬁrms (A, B and C) with probabilities 0.4, 0.35 and 0.25 respectively. From pas

t experience it is known that the probability of cost overruns for the ﬁrms are 0.05, 0.03, and 0.15, respectively.
a. Suppose a cost overrun is experienced by the company. What is the probability that the consulting ﬁrm involved is company A?

b. Are the events selecting company A, and incurring in cost overruns independent?

Mathematics

1 answer:

Zina [12.3K] · Answer 1 · 2026-03-08T09:17:18+03:00

Answer:

(a) 0.29412.

(b) No, the selection of company A and the occurrence of cost overruns are not independent.

Step-by-step explanation:

We know that a certain federal agency uses three consulting firms: A, B, and C, with probabilities of 0.4, 0.35, and 0.25, respectively. Thus:

P(A) = 0.4, P(B) = 0.35, P(C) = 0.25.

From previous data, the probability of cost overruns for each firm is known as 0.05 (A), 0.03 (B), and 0.15 (C), meaning:

Let CO = Event of cost overruns.

P(CO/A) = 0.05, which represents the probability of overruns when firm A is involved.

Likewise, P(CO/B) = 0.03 and P(CO/C) = 0.15.

(a) The probability that A is the consulting firm leading to a cost overrun is P(A/CO):

We will apply Bayes' theorem to find this probability:

P(A/CO) = $\frac{P(A)*P(CO/A)}{P(A)*P(CO/A) + P(B)*P(CO/B) + P(C)*P(CO/C)}$ .

Thus:

= $\frac{0.4*0.05}{0.4*0.05 + 0.35*0.03 + 0.25*0.15}$ = 0.29412.

(b) No, choosing company A and experiencing cost overruns aren't independent, as overruns are associated with the consulting firm’s involvement, and the nature of overruns changes with different firms.