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) = 0.23, P(A2) = 0.26, P(A3) = 0.28,P(A1 A2) = 0.11, P(A1 A3) = 0.05, P(A2 A3) = 0.07, 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) = ___________

Explain this probability in words.

If the firm is awarded project 2, this is the chance they will also be awarded project 1.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.

(b)    P(A2 A3 | A1) = _______________

Explain this probability in words.

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

(c)    P(A2 A3 | A1) = _______________

Explain this probability in words.

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

Explain this probability in words.

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

Explanation / Answer

a)(a)    P(A2 | A1) = P(A1nA2)/P(A1) =0.11/0.23=0.4783

If the firm is awarded project 1, this is the chance they will also be awarded project 2.

b)P(A2 A3 | A1) = P(A1nA2nA3)/P(A1) =0.02/0.23=0.0870

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

c) P(A2 A3 | A1) = P((A1nA2)U(A1nA3))/P(A1)=(0.11+0.05-0.02)/0.23=0.6087

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(A1UA2UA3) =P(A1)+P(A2)+P(A3)-P(A1nA2)-P(A1nA3)-P(A2nA3)+P(A1nA2An3)

=0.23+0.26+0.28-0.11-0.05-0.07+0.02=0.56

P(A1 A2 A3 | A1 A2 A3) = 0.02/0.56=0.0357

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

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