2.08 - We have already seen The Multiplicative Law of Probability P(A B = P(B P(A|B provided P(B 0 P(A B = P(A P(B|A provided P(A 0 P(A B = P(A P(B

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1 1 Section 2.8 Some Probability Laws © John J Currano, 01/26/2010 2 We have already seen: The Multiplicative Law of Probability . P ( A B ) = P ( B ) P ( A | B ) provided P ( B ) 0 P ( A B ) = P ( A ) P ( B | A ) provided P ( A ) 0 P ( A B ) = P ( A ) P ( B ) if A and B are independent . Countable Additivity . If { A k | k K } is a countable collection of mutually exclusive events, then If A and B are mutually exclusive , then P ( A B ) = P ( A ) + P ( B ). If A , B , C are mut. exclusive , P ( A B C )= P ( A )+ P ( B P ( C ). Etc. = K k k K P ( A k ). A k P U 3 The Multiplicative Law of Probability generalizes to three or more events. Assume all have nonzero probability. Then: P ( A B ) = P ( A ) P ( B | A ) P ( A B C ) = P ( A ) P ( B | A ) P ( C | A B ) P ( A B C D ) = P ( A ) P ( B | A ) P ( C | A B ) P ( D | A B C ) etc. 4 () = K k k K k k A P A P U Consequences: If A and B are events: P ( A ) = P ( A – B ) + P ( A B ) P ( B ) = P ( A B ) + P ( B – A ) P ( A B ) = P ( A – B ) + P ( A B ) + P ( B – A ) Sketch of Proof : The results follow by countable additivity. As can be seen from the picture, A – B , A B , and B – A a re pairwise disjoint (mutually exclusive) and A = ( A – B ) ( A B ), B = ( A B ) ( B – A ), ( A B ) = ( A – B ) ( A B ) ( B – A ). Countable Additivity If { A k } countable and mutually exclusive, then
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2 5 Consequences: If A and B are events: P ( A ) = P ( A – B ) + P ( A B ) P ( B ) = P ( A B ) + P ( B – A ) P ( A B ) = P ( A – B ) + P ( A B ) + P ( B – A ) Additive Law of Probability P ( A B ) = P ( A ) + P ( B ) – P ( A B ). Non-Decreasing If A B , then P ( A ) P ( B ). -- Note that B = A ( B – A ), so P ( B ) = P ( A ) + P ( B – A ) P ( A ) Complement : P ( A ) = 1 – P ( A ) -- Note that S = A A so 1 = P ( S ) = P ( A ) + P ( A ) () = K k k K k k A P A P U Countable Additivity If { A k } countable and mutually exclusive, then 6 Additive Law of Probability : P ( A B ) = P ( A ) + P ( B ) – P ( A B ). P ( A ) = P ( A – B ) + P ( A B ) P ( B ) = P ( A B ) + P ( B – A ) P ( A B ) = P ( A – B ) + P ( A B ) + P ( B – A ) Complement : P ( A ) = 1 – P ( A ) Show: 1. P ( B – A ) = P ( B ) –P ( A B ) 2. If A B , then P ( B – A ) = P ( B ) ( A ) 3. P ( ) = 0 7 Inclusion-Exclusion P ( A B C )= P ( A ) + P ( B ) + P ( C ) P ( A B ) – P ( A C ) – P ( B C ) + P ( A B C ) Sketch of proof : A B C = A ( B C ) associative law A ( B C ) = ( A B ) ( A C ) distributive law P ( A ( B C )) = P ( A ) + P ( B C ) P ( A ( B C )) = P ( A ) + P ( B C ) – P (( A B ) ( A C )) Additive Law of Probability : P ( A B ) = P ( A ) + P ( B ) – P ( A B ) Text, p. 58 8 =
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This note was uploaded on 02/23/2011 for the course MATH 444 taught by Professor Any during the Fall '10 term at Roosevelt.

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2.08 - We have already seen The Multiplicative Law of Probability P(A B = P(B P(A|B provided P(B 0 P(A B = P(A P(B|A provided P(A 0 P(A B = P(A P(B

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