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ics141-lecture25-Probability

ics141-lecture25-Probability - University of Hawaii ICS141...

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25-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga
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25-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 25 Chapter 6. Discrete Probability 6.1 An Introduction to Discrete Probability 6.2 Probability Theory
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25-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of “Discrete Mathematics and Its Applications” written by Kenneth H. Rosen. Some slides were done by Prof. Baek
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25-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Chapter 6: Discrete Probability
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25-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Why Probability? In the real world, we often don’t know whether a given proposition is true or false. Probability theory gives us a way to reason about propositions whose truth is uncertain . It is useful in weighing evidence, diagnosing problems, and analyzing situations whose exact details are unknown.
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25-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Experiments & Sample Spaces A (stochastic) experiment is any process that yields one of a given set of possible outcomes and the outcome is not necessarily known in advance. The sample space S of the experiment is just the set of all possible outcomes. The outcome of an experiment is the specific point in a sample space.
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25-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Events An event E is any set of possible outcomes in S . That is, E S. E.g., the event that “less than 10 people show up for our next class” is represented as the set {0, 1, 2, …, 9} of values of # of people here next class. We say that event E occurs when the actual outcome o is in E , which may be written o E . Note that o E denotes the proposition (of uncertain truth) asserting that the actual outcome will be one of the outcomes in the set E .
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25-8 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Uniform Probability Measure If S is a finite sample space of equally likely outcomes , and E is an event, that is, a subset of S , then the probability of E is p ( E ) = | E | / | S | Example: Coin toss Sample space: S = {H, T} Events E = , then p ( E ) = 0 E = {H}, then p ( E ) = 1/2 E = {T}, then p ( E ) = 1/2 E = {H, T}, then p ( E ) = 1
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25-9 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Example: Balls-and-Urn Suppose an urn contains 4 blue balls and 5 red balls.
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