Lecture 02-2007 - Lecture II Using the example from...

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Lecture II
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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 first 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 different sets of numbers can be  drawn? First, we note that we could draw any one of 50  numbers in the first draw.
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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 different ways  to draw two numbers 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 ( 29 ( 29 50 5 50 1 50 6 1 45 1 50! 50 50 6 ! k j k k k j k k = - = = = - = = = - U
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Finally, note that there are 6! ways to draw a set of 6  numbers (you could draw 35 first, or 20 first, …). Thus,  the total number of ways to draw an unordered set of 6  numbers out of 50 is This is a combinatorial. It also is useful for binomial  arithmetic: ( 29 50 50! 15,890,700 6 6! 50 6 ! � � = = � � - � � ( 29 0 n n k n k k n a b a b k - = �� + = �� �� U
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Basic definitions: 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. 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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Simple Event : An event which cannot be a union  of other events 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}. Composite Event : An event which is not a simple  event.
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A set  of different combinations of outcomes is  called an event. These events could be  simple events or compound events. In the  Texas lotto case, the important aspect is  that the event is something you could bet  on (for example, you could bet on three  numbers in the draw {35,20,15}). { } 1 , k j j ϖ ϖ K
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A collection of events  F  is called a family  of subsets of sample space  Ω . This family  consists of all possible subsets of  Ω   including  Ω  itself and the null-set  Φ .
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