Unformatted text preview: i ) n (A n B j) = 0 for i =1= j. 31. (a) [BB] Using Proposition 6.1.1(c), we have P(A B) = IA" BI = IAI IA n BI = ~ _ IA n BI = P(A) _ P(A B) " lSI lSI lSI lSI n . (b) From (a), P(A" B) = P(A) P(A n B) = P(A) (P(A) + P(B) P(A U B)) = P(A U B) P(B). (c) Using Proposition 6.1.1(d), we have P(A B) = IAEBBI = IAuBIIAnBI = IAUB _IAnBI EB lSI lSI lSI lSI = P(A U B) P(A n B) = (P(A) + P(B) P(A n B)) P(A n B) = P(A) + P(B) 2P(A n B). (d) [BB] From (c), P(A EB B) P(A) + P(B) 2P(A n B) = P(A) + P(B) 2(P(A) + P(B) P(A U B)) = 2P(A U B) P(A) P(B). (e) [BB] P(A n (B U e)) = P((A n B) U (A n e)) = P(A n B) + P(A n e) P(A n B n e). (t) Using (e), P(A n (B U e)) = (P(A) + P(B) P(A U B)) + (P(A) + p(e) P(A U e)) P(A n B n e) = 2P(A) + P(B) + p(e) P(A U B) P(A U e) P(A n B n e). (g) P(A U (B n e) P((A U B) n (A U e)) = P(A U B) + P(A U e) P(A U B U e)....
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 Summer '10
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 Graph Theory, Mutually Exclusive, Ordered pair, LSI

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