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# Ex07 - Review of Chapter 5 Review The Addition Principle...

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Review of Chapter 5

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The Addition Principle r 1 different objects in the first set, r 2 different objects in the second set, …, r m different objects in the m th set, # of ways to select an object form one of the m sets: r 1 + r 2 +…+ r m . disjoint AMS301, Summer 2009, Ning SUN
The Multiplication Principle A procedure -> m successive (ordered) stages: r 1 different outcomes in the first stage, r 2 different outcomes in the second stage, …, r m different outcomes in the m th stage. # of different composite outcomes the total procedure: r 1 r 2 r m . Independent distinct AMS301, Summer 2009, Ning SUN

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Summary Arrangement (ordered outcome) or Distribution of distinct objects Selection (unordered outcome) or Distribution of identical objects No Repetition P ( n , r ) C ( n , r ) Unlimited Repetition n r C ( r + n -1, r ) Restricted Repetition P ( r; r 1 , r 2 , …, r n ) ---- AMS301, Summer 2009, Ning SUN
Basic Formulas ! ! ) , ( k n k n P = )! ( ! ! ) , ( k n k n k n k n C - = = ! !... ! ! ) ,..., , ; ( 2 1 2 1 n n r r r r r r r r P = AMS301, Summer 2009, Ning SUN

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Integer Integer-solution solution-of of-an an-equation version equation version 1. The number of ways to select r objects with repetition from n different types of objects. 2. The number of ways to distribute r identical objects into n distinct boxes. 3. The number of nonnegative integer solutions to x 1 + x 2 +…+ x n = r . 12 = 4 + 3 + 1 + 4 AMS301, Summer 2009, Ning SUN
Probability Probability = # desired outcomes # total outcomes AMS301, Summer 2009, Ning SUN

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5.2#16 What is the probability that a five-card poker hand has the following? (b) four of a kind ( 9 9 9 9 J ) (c) Two pairs ( J J 4 4 9 ) (d) A full house ( 3 3 3 6 6 ) Sol: hw3 AMS301, Summer 2009, Ning SUN
Sample #2 What is the probability that a 7-card poker hand chosen from the 52 cards in a deck has exactly 3 pairs (no 3-of-a-kind or 4-of-a-kind)?

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