# Since the two s t paths involve separate sets of

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Unformatted text preview: of complex operations. A common theme is to try to evaluate the complexity of a large system by recursively evaluating the reliability of its subsystems. Often no more underlying probability theory is required beyond that covered earlier in this chapter. However, intuition can be sharpened by considering the case that many of the events have very small probabilities. 2.12.1 Union bound A general tool for bounding failure probabilities is the following. Given two events A and B , the union bound is P (A ∪ B ) ≤ P (A) + P (B ). A proof of the bound is that P (A) + P (B ) − P (A ∪ B ) = P (AB ) ≥ 0. If the bound, P (A) + P (B ), is used as an approximation to P (A∪B ), the error or gap is P (AB ). If A and B have large probabilities, the gap can be signiﬁcant; P (A) + P (B ) might even be larger than one. However, in general, P (AB ) ≤ min{P (A), P (B )}, so the bound is never larger than two times P (A ∪ B ). Better yet, if A and B are independent and have small probabilities, then P (AB ) = P (A)P (B ) &lt;&lt; P (A ∪ B ) (h...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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