Question

A consulting firm submitted a bid for a large research project. The firm's management initially felt that 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 35% 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.
Thanks to whoever can help with this one!

Answer 1

Answer:

• a) 0.500

• b) 0.350

• c) 0.318

Explanation:

A probability tree diagram is very helpful, almost necessary, to work this kind of problems.

Let's us simulate a probability tree diagram:

• Successful bid: 0.5

                      Request additional information: 0.75 × 0.5 = 0.375                              

                      No request                                : 0.25 × 0.5 = 0.125

• Failed bid: 0.5

                      Request additional information: 0.35 × 0.50 = 0.175

                      No request                                 : 0.65 × 0.50 = 0.325

Call S the event of a succesful bid  and R the event of requestion additional information. Thus,

• P(S) is the probability of a succesfull,

• P(R) is the probability of requesting additional information, and

• P(R∩S) = p(S∩R) is the joint probability of a succesful bid and requested information.

Questions

a. What is the prior probability of the bid being successful(that is, prior to the request for additional information)

It is P(S). It is the 0.500 because it is said that there is a 50-50 chance, thus P(S) = 50/100 = 0.500.

• P(S) = 0.500

b. What is the conditional probability of a request for additional information given that the bid will ultimately be successful?

You want P(R/S).

Then, you can use the formula for conditional probability, which is:

• P(R/S) = P(R∩S) / P(S)

You need to determine P(R∩S). This is the probability of a being succesful and addtional information is requested.

You can take it directly from the corresponding branch of your probabiity tree: it is P(S∩R) = 0.35 × 0.50 = 0.175

From the first question, P(S) = 0.500, then:

• P(R/S) = P(R∩S) / P(S) = 0.175 / 0.50 = 0.350

c. Compute the posterior probability that the bid will be successful given a request for additional information.

Now you want P(S/R).

That is:

• P(S/R) = P(S∩R)/P(R)

P(R) must be taken from the tree diagram: 0.375 + 0.175 = 0.55

You already have P(S∩R) from the previous question. It is 0.175

Thus, substituting:

• P(S/R) = P(S∩R)/P(R) = 0.175 / 0.55 = 0.318