18_05_lec5

# 18_05_lec5 - 18.05 Lecture 5 Â 2.2 Independence of events P...

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Unformatted text preview: 18.05 Lecture 5 February 14, 2005 Â§ 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 Ã— Independence does not imply that the sets do not intersect. Disjoint = Independent. âˆˆ 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 ) âˆ’ P ( AB ) = P ( A ) âˆ’ P ( A ) P ( B ) = P ( A )(1 âˆ’ 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....
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18_05_lec5 - 18.05 Lecture 5 Â 2.2 Independence of events P...

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