Discrete Mathematics with Graph Theory (3rd Edition) 199

# Discrete Mathematics with Graph Theory (3rd Edition) 199 -...

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Section 7.4 197 28. We use induction on n, the strong form. If n = 2, the right hand side of the equation is just P(A 1 )P(A 2 1 AI) = P(A 1) P~(A~:1 = P(A 2 n AI), which is the left hand side, so the state- ment is true. Now assume n > 2 and the result holds for all k, 1 :::; k < n. We have P(A 1 n A2 n ... nAn) = P((A 1 n A2 n··· nAn-I) nAn) = P(A 1 n A2 n ... n An-I)P(A n I Al n A2 n ... n An-I) by the k = 2 case = [P(A1 )P(A 2 1 AI) ... P(A n- 1 I Al n A2 n ... n An- 2 )] P(A n I Al n A2 n ... nAn-I) by the k = n - 1 case, and this is the right hand side. By the Principle of Mathematical Induction, the result follows. 29. [BB] Let A, B, C be the events of eating at Leonce's, Harold's, and Vegan Delights, respectively, and let D be the event a customer is satisfied. We are given that P(A) = .4, P(B) = .35, P(C) = .25, P(D I A) = .85, P(D I B) = .9, P(D I C) = .95. (a) P(D n C) = P(C)P(D I C) = .25(.95) = .2375. (b) P(B n DC) = P(B) - P(B n D) = P(B) - P(B)P(D I B) = .35(1 - .9) = .035. (c) P(D) = P(A)P(D I A) + P(B)P(D I B) + P(C)P(D I C) = .4(.85) + .35(.9) + .25(.95) = .8925. P(A)P(D I A) .4(.85) . (d)
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