Lecture02-2010 - Basic Axioms of Probability: Lecture II...

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Basic Axioms of Probability: Lecture II Charles B. Moss May 7, 2010 I. Basics of Probability A. Using the example from Birenens Chapter 1: Assume we are in- terested in the game Texas lotto (similar to Florida lotto). 1. 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. 2. How many di²erent sets of numbers can be drawn? a. First, we note that we could draw any one of 50 numbers in the ±rst draw. b. 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 c. Again, for the third draw, we only have 48 possible num- bers left. Therefore, the total number of possible ways to choose6numbersouto f50is 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) d. 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 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture II Fall 2010 50 6 ! = 50! 6! (50 6)! =15 , 890 , 700 (2) e. This is a combinatorial. It also is useful for binomial arithmetic: ( a + b ) n = n X k =0 n k ! a k b n k (3) B. De±nitions 1. 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. 2. Event : A subset of the sample space. In the Texas lotto sce- nario, 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,. 3. Simple Event : An event which cannot be a union of other events. In the Texas lotto scenario, this is a single draw such as 35,20,15,1,5,45.
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture02-2010 - Basic Axioms of Probability: Lecture II...

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