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Unformatted text preview: Laws of Operation: Identity - ∅ = ∅ ∩ = ∪ = ∩ = ∅ ∪ A U U A A U A A A , , Complement - A A A A U A A = ∅ = ∩ = ∪ , , Commutative - A B B A A B B A ∩ = ∩ ∪ = ∪ , DeMorgan’s - B A B A B A B A ∪ = ∩ ∩ = ∪ , Associative - ( 29 ( 29 ( 29 ( 29 C B A C B A C B A C B A ∩ ∩ = ∩ ∩ ∪ ∪ = ∪ ∪ , Distributive - ) ( ) ( ) ( ), ( ) ( ) ( C A B A C B A C A B A C B A ∩ ∪ ∩ = ∪ ∩ ∪ ∩ ∪ = ∩ ∪ elementsS elementsA S A A # # ) Pr( = = Probability Theorems If ø is the empty set then P(ø)=0 ) Pr( 1 ) Pr( A A- = For any 2 events: ) Pr( ) Pr( ) Pr( ) Pr( B A B A B A ∩- + = ∪ For any 3 events: ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( C B A C B C A B A C B A C B A ∩ ∩ + ∩- ∩- ∩- + + = ∪ ∪ Finite Sample Spaces Permutations – an ordered arrangement of distinct objects. One permutation differs from another if the order of arrangement differs or if the content differs. Definitions: there are n distinct objects of which we wish to choose permutations of r objects (r ≤ n) the number of permutations, P n r, is given by: ) 1 )...( 3 )( 2 )( 1 ( +---- r n n n n n P n r *note: ! n P n n = and 0!=1 )! ( ! , r n n r n P- = n - # of things r - # of things taken at a time *concerned with order ) , , ( ) , , ( c a b c b a ≠ Combinations – an arrangement of distinct objects where one combination differs from another ONLY IF the content of the arrangement offers. ORDER does not matter. Definition: there are n distinct objects of which we wish to choose combinations of r objects. ! )! ( ! r r n n r n- = Relationship between Combinations and Permutation * a permutation is the selection of r objects from the n and then permute the r objects )! ( ! ! r n n r n r n r P- = = Hypergeometric Sampling – the population has N items of which D belong to same class of interest. A random sample of size n is selected without replacement. If A denotes the event of obtaining exactly r items from the class of interest in the sample, then… -- = n N r n D N r D A P ) ( Multinomial Sampling Problem- ! !... ! ! 2 1 ... , 2 1 x n n n n n n n n P x = Ex) TENNESSEE *the number of ways to arrange 9 letters Conditional Probability – the probability of the events where the event is conditioned on some subset of the sample space. Definition: the conditional probability of event A given event B ) Pr( ) Pr( ) Pr( B B A B B A B A ∩ = ∩ = if P(B)>0 Multiplication Rule ) ( ), ( ) ( ) ( ) ( ), ( ) ( ) ( ⋅ = ∩ ⋅ = ∩ A P A B P A P B A P B P B A P B P B A P Independence – if the probability of the occurrence of one is not affected by the occurrence or nonoccurrence of the other....
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This note was uploaded on 04/14/2008 for the course ISYE 3770 taught by Professor Goldsman during the Fall '07 term at Georgia Tech.
- Fall '07