# 11_mincut - Chapter 11 Min Cut By Sariel Har-Peled October 1 2007x To acknowledge the corn This purely American expression means to admit the

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11.1.2 Some Deﬁnitions We remind the reader of the following concepts. The conditional probability of X given Y is Pr ± X = x | Y = y ² = Pr ± ( X = x ) ( Y = y ) ² / Pr ± Y = y ² . An equivalent, useful restatement of this is that Pr ± ( X = x ) ( Y = y ) ² = Pr ³ X = x ´ ´ ´ ´ Y = y µ · Pr ± Y = y ² . (11.1) Two events X and Y are independent , if Pr ± X = x Y = y ² = Pr [ X = x ] · Pr ± Y = y ² . In particular, if X and Y are independent, then Pr ³ X = x ´ ´ ´ ´ Y = y µ = Pr [ X = x ]. The following is easy to prove by induction using Eq. (11.1). Lemma 11.1.2 Let E 1 , . . . , E n be n events which are not necessarily independent. Then, Pr ± n i = 1 E i ² = Pr [ E 1 ] * Pr [ E 2 |E 1 ] * Pr [ E 3 |E 1 ∩ E 2 ] * . . . * Pr ³ E n ´ ´ ´ ´ E 1 . . . ∩ E n - 1 µ .
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## This note was uploaded on 06/14/2009 for the course CS 473 taught by Professor Viswanathan during the Fall '08 term at University of Illinois at Urbana–Champaign.

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11_mincut - Chapter 11 Min Cut By Sariel Har-Peled October 1 2007x To acknowledge the corn This purely American expression means to admit the

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