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# 5 - 18.05 Lecture 5 2.2 Independence of events P(A|B = P(AB...

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18.05 Lecture 5 February 14, 2005 § 2.2 Independence of events. P ( A B ) = P ( AB ) P ( B ) ; | Definition - A and B are independent if P ( A B ) = P ( A ) | P ( AB ) P ( A B ) = = P ( A ) P ( AB ) = P ( A ) P ( B ) | P ( B ) Experiments can be physically independent (roll 1 die, then roll another die), or seem physically related and still be independent. Example: A P ( A 1 ) = { } . P ( B 2 { 1 , 3 } = P ( AB 1 3 odd , B ) = ) = , therefore independent. = = { 1, 2, 3, 4 } . Related events, but independent. .AB = 2 3 2 P ( AB ) = 1 3 2 × Independence does not imply that the sets do not intersect. Disjoint = Independent. If A, B are independent, find P ( AB c ) P ( AB ) = P ( A ) P ( B ) AB c = A \ AB , as shown: so, P ( AB c ) = P ( A ) - P ( AB ) = P ( A ) - P ( A ) P ( B ) = P ( A )(1 - P ( B )) = P ( A ) P ( B c ) therefore, A and B c are independent as well. similarly, A c and B c are independent. See Pset 3 for proof. Independence allows you to find P (intersection) through simple multiplication. 14

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2 Example: Toss an unfair coin twice, these are independent events. P ( H ) = p, 0 p 1 , find P (“ T H ) = tails first, heads second P (“ T H ) = P ( T ) P ( H ) = (1 - p ) p 1 Since this is an unfair coin, the probability is not just 4 T H 1 = If
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5 - 18.05 Lecture 5 2.2 Independence of events P(A|B = P(AB...

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