Slides02-2010 - Basic Axioms of Probability: Lecture II...

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Basic Axioms of Probability: Lecture II Charles B. Moss August 23, 2010 Charles B. Moss () Basic Axioms of Probability: Lecture II August 23, 2010 1 / 20
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Outline 1 Basic Probability Theory Course Objectives Defnitions Axiomatic Foundations Axioms o± Probability Counting Techniques Charles B. Moss () Basic Axioms of Probability: Lecture II August 23, 2010 2 / 20
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Basic Probability Theory Course Objectives Basics of Probability Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto). In this game, players choose a set of 6 numbers out of the ±rst 50. Note that the ordering does not count so that 35,20,15,1,5,45 is the same of 35,5,15,20,1,45. How many di²erent sets of numbers can be drawn? First, we note that we could draw any one of 50 numbers in the ±rst draw. However for the second draw we can only draw 49 possible numbers (one of the numbers has been eliminated). Thus, there are 50 x 49 di²erent ways to draw two numbers Charles B. Moss () Basic Axioms of Probability: Lecture II August 23, 2010 3 / 20
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Basic Probability Theory Course Objectives Again, for the third draw, we only have 48 possible numbers left. Therefore, the total number of possible ways to choose 6 numbers out of 50 is 5 Y j =1 (50 j )= 50 Y k =45 k = 50 Q k =1 k 50 6 Q k =1 k = 50! (50 6)! (1) Finally, note that there are 6! ways to draw a set of 6 numbers (you could draw 35 ±rst, or 20 ±rst, ). Thus, the total number of ways to draw an unordered set of 6 numbers out of 50 is ± 50 6 ² = 50! 6! (50 6)! =15 , 890 , 700 (2) Charles B. Moss () Basic Axioms of Probability: Lecture II August 23, 2010 4 / 20
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Basic Probability Theory Course Objectives This is a combinatorial. It also is useful for binomial arithmetic: ( a + b ) n = n X k =0 ± n k ² a k b n k (3) Charles B. Moss () Basic Axioms of Probability: Lecture II August 23, 2010 5 / 20
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Basic Probability Theory Defnitions Defnitions Sample Space : The set of all possible outcomes. In the Texas lotto scenario, the sample space is all possible 15,890,700 sets of 6 numbers which could be drawn. Event : A subset of the sample space. In the Texas lotto scenario, possible events include single draws such as 35,20,15,1,5,45 or complex draws such as all possible lotto tickets including 35,20,15. Note that this could be 35,20,15,1,2,3, 35,20,15,1,2,4,.
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Slides02-2010 - Basic Axioms of Probability: Lecture II...

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