A consulting firm submitted a bid for a large research project. The firm’s management
initially felt they had a 50–50 chance of getting the project. However, the agency to which
the bid was submitted subsequently requested additional information on the bid. Past
experience indicates that for 75% of the successful bids and 40% of the unsuccessful bids
the agency requested additional information.

a. What is the prior probability of the bid being successful (that is, prior to the request
for additional information)?
b. What is the conditional probability of a request for additional information given that
the bid will ultimately be successful?
c. Compute the posterior probability that the bid will be successful given a request for
additional information.

(a) 0.50

(b) 0.75

(c) 0.6522

Step-by-step explanation:

We are given that the firm’s management  initially had a 50–50 chance of getting the project.

Let Probability of getting a project or bid being successful, P(S) = 0.50

Probability of not getting a project or bid being unsuccessful, P(US) = 1 - 0.50 = 0.50

Also, Past  experience indicates that for 75% of the successful bids and 40% of the unsuccessful bids  the agency requested additional information which means;

Let event R = agency requested additional information

So, Probability that the agency requested additional information given the bid was successful, P(R/S) = 0.75

Probability that the agency requested additional information given the bid was unsuccessful, P(R/US) = 0.40

(a) Prior probability of the bid being successful = Probability of getting a project or bid being successful = = 0.50

(b) The conditional probability of a request for additional information given that  the bid will ultimately be successful = P(R/S) = 0.75

(c) The posterior probability that the bid will be successful given a request for  additional information is given by P(S/R) ;

Using Bayes' Theorem for this we get;

   P(S/R) = = = 0.6522 .