Independent Pr[A] = Pr[A | B] Definition 2: Events Events A and B - PowerPoint PPT Presentation

Independent Events Mathematics for Computer Science MIT 6.042J/18.062J Definition 1: Events A and B are independent iff Independent Pr[A] = Pr[A | B] Definition 2: Events Events A and B are independent iff Pr[A] ⋅ Pr[B] = Pr[A ∩ B] Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 indep inde -events.1 p-events.1 indep inde -events.2 p-events.2 Definitions of Independence Definitions of Independence proof of equivalence: need Pr[B] ≠ 0 for Def. 1. iff Pr[A] = Pr[A | B] Def. 2 always works: Pr[A ∩ ∩ B] iff Pr[A] = Pr[A} ⋅ Pr[B] = Pr[A ∩ B] Pr[B] Pr[A] ⋅ Pr[B] = Pr[A ∩ B] Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 indep indep-events.3 -events.3 indep indep-events.4 -events.4 1

Independence Independence Pr[A] ⋅ Pr[B] = Pr[A ∩ B] Corollary: If Pr[B]= 0, then symmetric in A and B so, B is independent of every A independent of B iff event B independent of A Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 indep-events.5 indep-events.5 indep-events.6 indep-events.6 Independence Independence Corollary: If Pr[B]= 0, then A independent of B B is independent of every means event – even itself. Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 indep-events.7 indep-events.7 indep-events.9 indep-events.9 2

Independence Independence Lemma: A independent of B A independent of B iff means A is independent of B A independent of . whether or not B occurs: Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 indep-events.10 indep-events.10 indep-events.11 indep-events.11 Independence Lemma: A independent of B iff B A independent of Simple proof using: Pr[A - B] = Pr[A] - Pr[A ∩ B] Albert R Meyer, May 3, 2013 indep-events.12 indep-events.12 3

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