A computer consulting firm presently has bids out on three projects. Let A_1 = (

Question

A computer consulting firm presently has bids out on three projects. Let A_1 = (awarded project i), for i = 1 2, 3, and suppose that P(A_1) = 0.22, P(A_2) = 0.25, P(A_3) = 0.29, P(A_1 intersection A_2) = 0.07, P(A_1 intersection A_3) = 0.08, P(A_2 intersection A_3) = 0.09, P(A_1 intersection A_2 intersection A_3) = 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(A_2 | A_1) = Explain this probability in words. If the firm is awarded project 2, 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 project 2. This is the probability that the firm is awarded either project 1 or project 2. This is the probability that the firm is awarded both project 1 and project 2. P(A_2 intersection A_3 | A_1) = Explain this probability in words. If the firm is awarded 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 projects 2 and 3. This is the probability that the firm is awarded projects 1, 2, and 3. (c) p(A_2 union A_1) = 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 the projects 2 and 3, this is the chance they will also her 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. This is the probability that the firm is awarded at least one of the projects. (d) p(A_1 intersection A_2 intersection A_3 | A_1 union A_2 union A_3) = 0.037

Explanation / Answer

(a) P(A1) = 0.22 P(A2) = 0.25 P(A1 A2) = 0.07

=> P(A2|A1) = P(A1 A2) / P(A1) = 0.07 / 0.22 = 0.3182

(Note: The question asks to round to FOUR decimal places!)

In words : If the firm is awarded project 1, this is the chance they will be awarded project 2.

(b) P(A2 A3 | A1) = P(A2 A3 A1) / P (A1) = 0.02 / 0.22 = 0.0909

In words : If the firm is awarded project 1, this is the chance they will be awarded projects 2 and 3.

(c) P(A2U A3) = P(A2) + P(A3) - P(A2 A3) = 0.25 + 0.29 - 0.09 = 0.45

P(A2U A3 U A1) = 0.22 + 0.25 + 0.29 - 0.07 - 0.08 - 0.09 + 0.02 = 0.54

P(A2U A3 A1) = P(A2U A3) + P(A1) - P(A2U A3 U A1) = 0.45 + 0.22 - 0.54 = 0.13

P(A2U A3 | A1) = P(A2U A3 A1) / P (A1) = 0.13 / 0.22 = 0.5909

In words : If the firm is awarded project 1, this is the chance they will be awarded atleast one of the other two projects.

(d) P(A2 A3 A1 | A2U A3 A1) = 0.037

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