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lecture2_updated - OR Job 2 in p q ways • “OR”...

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7/26/11 Lecture 2 4th May 2011
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7/26/11 Recall: Definitions of Probability: 1. Classical definition 2. Relative Frequency definition 3. Subjective Probability definition •. Sample space (S): discrete vs. non- discrete Simple vs. compound Probability distribution on S
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7/26/11 Prob 2.1: Students in a particular program have the same 4 math profs. Two students in the program each independently ask one of their math profs for a letter of reference. Assume each is equally likely to ask any of the math profs. a) List a sample space for this “experiment”.
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7/26/11 b) Find the probability both students ask the same prof.
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7/26/11 Probability: Counting Techniques Recall: if we specify a sample space S={a1, a2, …, an}, where each simple event has probability 1/n (i.e. are equally likely) Then defining the compound event A which contains r points we have P(A)= r/n. To define more complicated
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7/26/11 Counting Arguments: 1. Addition Rule: Suppose Job 1 can be done in p ways and job 2 can be done in
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Unformatted text preview: OR Job 2, in p+q ways. •. “OR”: interpreted as addition 7/26/11 2. The Multiplication Rule: Suppose Job 1 can be done in p ways and, for each of these ways, Job 2 can be done q ways. Then we can do BOTH Job 1 AND Job 2 in pxq ways. • “AND”: interpreted as multiplication 7/26/11 7/26/11 7/26/11 7/26/11 • In a lot of problems, the sample space of interest consists of a set of arrangements or sequences. These are commonly called permutations . • Example: Arranging letters 7/26/11 7/26/11 • 7/26/11 • n 1 2 3 4 5 6 7 8 9 10 n! 1 1 2 6 24 12 72 504 4032 3628 80 36288 00 7/26/11 • 7/26/11 • As we just saw when n gets reasonably large, for example, when sampling from a deck of cards or a large population, we can no longer just count the number of cases we are interested in. • And the counting rules and arrangements can be used to simplify the situation. 7/26/11 • Examples:...
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