Summary of Probability Rules

Summary of Probability Rules - conditional probability P (...

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STAT 230 Summary of Probability Rules Basic Axioms: Let S be the sample space for an experiment. Then a probability rule P must satisfy: (i) P ( S )=1 (ii) 0 P ( A ) 1 for all A S (iii)* If A 1 ,A 2 ,...,A m ,... is a sequence (Fnite or countably inFnite) of disjoint events (i.e. A j A k = ), then P ( ± n A n )= ² n P ( A n ) (general additivity axiom) If S is Fnite, then (iii)* can be replaced by: (iii) If A 1 ,A 2 ,...,A m are disjoint events (i.e. A j A k = ), then P ( m ± n =1 A n )= m ² n =1 P ( A n ) (Fnite additivity axiom) Rules Which Follow From The Axioms: 1. If A B then P ( A ) P ( B ) and P ( B \ A )= P ( B ) - PA ). 2. P ( ¯ A )=1 - P ( A ); in particular P ( )=0. 3. ±or any sequence of events A 1 ,A 2 ,... (Fnite or countably inFnite), P ( ± n A n ) ² n P ( A n ) Equality holds if the events are disjoint. 4. P ( A B )= P ( A )+ P ( B ) - P ( A B ) for any events A and B . 5. P ( A B C )= P ( A )+ P ( B )+ P ( C ) - P ( A B ) - P ( A C ) - P ( B C )+ P ( A B C ). Conditional Probability, Independence, and Multiplication Rules:
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Unformatted text preview: conditional probability P ( A | B ) = P ( A B ) /P ( B ). basic multiplication rule P ( A B ) = P ( A ) P ( B | A ). generalized multiplication rule P ( A 1 A 2 . . . A m ) = P ( A 1 ) P ( A 2 | A 1 ) P ( A m | A 1 . . . A m-1 ). A and B are independent if P ( A B ) = P ( A ) P ( B ). If an experiment consists of a sequence of independent trials, and A 1 , . . ., A m are events such that A j depends only on the j th trial, then A 1 , . . ., A m are independent, and P ( A 1 A 2 . . . A m ) = P ( A 1 ) P ( A 2 ) P ( A m ) ....
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This note was uploaded on 10/21/2010 for the course STAT 230 taught by Professor Various during the Fall '06 term at Waterloo.

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