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LectureNotes2 - Foundations of Probability: Part II Cyr...

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Foundations of Probability: Part II Cyr Emile M’LAN, Ph.D. mlan@stat.uconn.edu Foundations of Probability: Part II – p. 1/25
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Counting Rules Text Reference : Introduction to Probability and Its Application, Chapter 2. Reading Assignment : Sections 2.4-2.5, January 28-February 2 When the various outcomes of an experiment are equally likely, the task of computing probabilities reduces to counting. However, when the number of simple events is large, this task of counting the number of simple events in the sample space and the event of interest becomes tedious or even impossible. Foundations of Probability: Part II – p. 2/25
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Counting Techniques Fundamental Principles of Counting Our first counting rule applies to any situation in which an event consists of ordered pairs of objects and we wish to count the number of such pairs. Theorem 2.2a : If the first task of an experiment can result in n 1 possible outcomes and, for each of these such outcomes, the second task can result in n 2 possible outcomes, then there are n 1 n 2 possible outcomes for the two tasks together. Example 2.7 : A homeowner doing some remodeling requires the services of both a plumbing contractor and an electrical contractor. If there are 12 plumbing contractors and 9 electrical contractors available in the area, in how many ways can the contractors be chosen? Foundations of Probability: Part II – p. 3/25
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Counting Techniques Solution : If we denote the plumbers by P 1 ,...,P 12 and the electricians by Q 1 ,...,Q 9 , then we wish the number of pairs of the form ( P i ,Q j ) . With n 1 = 12 and n 2 = 9 , the product rule yields N = (12)(9) = 108 possible ways of choosing the two types of contractors. Extension of the Fundamental Principles of Counting to k tasks Theorem 2.2b : Suppose an experiment consists of k ordered tasks and that the first task can result in n 1 possible outcomes; for each outcome of the first task, the second task can result in n 2 possible outcomes;. ..; for each possible outcome of the first k - 1 task, the k task can result in n k possible outcomes. Then there are n 1 n 2 ...n k possible k -tuples (outcomes of the k ordered tasks). Foundations of Probability: Part II – p. 4/25
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Example 2.8 : The fixed-price dinner at a local restaurant provides the following choices: Appetizer : seafood soup or salad or egg rolls or crab rangoon Entre : baked chicken or sweet and sour chicken or diced lemon chicken or broiled beef patty or teriyaki beef or beef with snow pea or beef boneless rib Desert : ice cream or cheesecake Solution : There are four choices of an appetizer. For each choice of appetizer, there 7 choices of entre, and that for each of the 4 · 7 = 28 choices, there are 2 choices for desert. Hence, a total of
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LectureNotes2 - Foundations of Probability: Part II Cyr...

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