In Exercise 13, is any ?A?i ?independent of any other ?A?j? Answer using the multiplication property for independent events. Reference exercise- 13 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) =.22, P(A2) = .25, P(A3) = .28, P(A1?A2) = .11,P?(A1?A3)= .05, P(A2?A3) = .07, P(A1?A2?A3)= .01. Express in words each of the following events, and compute the probability of each event: a. A1?A2 b.A?1?A’2[?Hint:(? A1?A2) ?= ?(? A?1?A?2] c.A1?A2?A3 d. A?1?A?2?A’3 e. A?1?A?2?A3 f.( A?1?A?2) ?A3

Answer : Step 1 of 6 : Given, A computer consulting firm presently has bids out on three projects. Let Ai = { awarded project i}, for i = 1,2,3, suppose that P(A1) =.22 P(A2) = .25 P(A3) = .28 P(A1A2) = .11 P(A1A3)= .05 P(A2 A3) = .07 P(A1A2A3)= .01. Probability of independence condition is P(AB) = P(A) P(B) a) P(A 1 ) =2P(A ) P(A1) 2 = (0.22) (0.25) = 0.055 Step 2 of 6 : ` ` b) P(A 1 ) = 21 - P ( A )) (11- P ( A )) 2 = (1 - 0.22)(1-0.25) = (0.78) (0.75) = 0.585 Step 3 of 6 : c) P(A 1A 2 A 3 = P(A ) 1 P(A ) +2P( A ) 3 - P( A1 A 2 - P( A 2 A3) - ( A1 A 3 + P( A1 A2 A3) P(A 1A 2 A 3 = 0.22 + 0.25 + 0.28 - 0.11 - 0.07 - 0.05 + 0.01 = 0.53