Bonferroni's inequality. (a) Prove that for any two events
Chapter , Problem 11(choose chapter or problem)
Bonferroni's inequality. (a) Prove that for any two events A and B, we have P(A n B) P(A) + P(B) - 1. (b) Generalize to the case of n events AI, A2, . .. , An, by showing that P(AI n A2 n n An) P(AI ) + P(A2) + . . . + P(An) - (n - 1). Solution. We have P(A U B) = P(A) + P(B) - P(A n B) and P(A U B) :::; 1. which implies part (a). For part (b), we use De Morgan's law to obtain 1 - P(AI n n An) = P((AI n n Anr) = P(A U U A) :::; P(A) + . . . + P(A) = (1 - P(Ad) + . . . + (1 - P(An)) = n - P(AI) - ... - P(An).
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