A computer consulting firm presently has bids out on three projects. Let A i = {

Question

A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.26, P(A3) = 0.29, P(A1 A2) = 0.08, P(A1 A3) = 0.07, P(A2 A3) = 0.11, P(A1 A2 A3) = 0.02. Use the probabilities given above to compute the following probabilities, and explain in words the meaning of each one. (Round your answers to four decimal places.)

(a)    P(A2 | A1) = 1

Explain this probability in words.

This is the probability that the firm is awarded both project 1 and project 2. This is the probability that the firm is awarded either project 1 or project 2.     If the firm is awarded project 1, this is the chance they will also be awarded project 2. If the firm is awarded project 2, this is the chance they will also be awarded project 1.

(b)    P(A2 A3 | A1) = 3

Explain this probability in words.

If the firm is awarded project 1, this is the chance they will also be awarded projects 2 and 3. This is the probability that the firm is awarded at least one of the projects.     This is the probability that the firm is awarded projects 1, 2, and 3. If the firm is awarded projects 2 and 3, this is the chance they will also be awarded project 1.

(c)    P(A2 A3 | A1) = 5

Explain this probability in words.

This is the probability that the firm is awarded projects 1, 2, and 3. If the firm is awarded at least one of projects 2 and 3, this is the chance they will also be awarded project 1.     This is the probability that the firm is awarded at least one of the projects. If the firm is awarded project 1, this is the chance they will also be awarded at least one of the other two projects.

(d)    P(A1 A2 A3 | A1 A2 A3) = 7

Explain this probability in words.

This is the probability that the firm is awarded projects 1, 2, and 3. If the firm is awarded at least two of the projects, this is the chance that they will be awarded all three projects.     This is the probability that the firm is awarded at least one of the projects. If the firm is awarded at least one of the projects, this is the chance that they will be awarded all three projects.

Explanation / Answer

Answers)

P(A1) = 0.22    P(A2) = 0.26    P(A3) = 0.29 P(A1 n A2) = 0.08 P(A1 n A3) = 0.07 P(A2 n A3) = 0.11 and

P(A1 n A2 n A3) = 0.02

(a) P(A2 / A1) = P(A2 n A1) / P(A1) = 0.08 / 0.22 = 0.3636.

    If the firm is awarded Project 1, this is the chance that they will also be awarded project 2 = 0.3636

(b) P(A2 n A3 / A1) = P[(A2 n A3) n A1 ] / P(A1) = P(A2 n A3 n A1) / P(A1) = 0.02 / 0.22 = 0.0909

If the firm is awarded project 1, this is the chance that they will also be awarded projects 2 and 3 = 0.0909

(c) P(A2 U A3 / A1) = P(A2/A1) + P(A3/A1) - P(A2 n A3 / A1)

                                = ( 0.08 / 0.22) + (0.07 / 0.22) - 0.0909 = 0.5909

If the firm is awarded project 1, this is the chance that they will also be awarded atleast one of the other two projects is 0.5909

(d) P(A1 n A2 n A3 / A1 U A2 U A3)

P(A1 U A2 U A3) = P(A1)+ P(A2)+ P(A3) P(A1 A2) P(A2 A3) P(A1 A3)+ P(A1 A2 A3)

                            = 0.22 + 0.26 + 0.29 - 0.08 - 0.11 - 0.07 + 0.02

                            = 0.53

Therefore, P(A1 n A2 n A3 / A1 U A2 U A3)   = (0.02*0.53) / 0.53 = 0.02

If the firm is awarded atleast one of the projects, this is the chance that they will be awarded all three projects.

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