Berkeley COMPSCI 70 - Lecture Notes (6 pages)

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Lecture Notes



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Lecture Notes

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Lecture Notes


Pages:
6
School:
University of California, Berkeley
Course:
Compsci 70 - Discrete Mathematics and Probability Theory
Discrete Mathematics and Probability Theory Documents

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CS 70 Fall 2003 Discrete Mathematics for CS Wagner Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability We will aim to make sense of statements such as the following 1 There is a 30 chance of a magnitude 8 earthquake in Northern California before 2030 2 The average time between system failures is about three days 3 The chance of getting a flush in a five card poker hand is about 2 in 1000 4 In this load balancing scheme the probability that any processor has to deal with more than twelve requests for service is negligible Implicit in all such statements is the notion of an underlying probability space This may be the result of some model we build of the real world as in 1 and 2 above or of a random experiment that we have ourselves constructed as in 3 and 4 above None of these statements makes sense unless we specify the probability space we are talking about for this reason statements like 1 which are typically made without this context are almost content free Probability spaces Every probability space is based on a random experiment such as rolling a die shuffling a deck of cards picking a number assigning jobs to processors running a system etc Rather than attempt to define experiment directly we shall define it by its set of possible outcomes which we call a sample space Note that all outcomes must be disjoint and they must cover all possibilities Definition 17 1 sample space The sample space of an experiment is the set of all possible outcomes An outcome is often called a sample point or atomic event Definition 17 2 probability space A probability space is a sample space together with a probability Pr for each sample point such that 0 Pr 1 for all Pr 1 i e the sum of the probabilities of all outcomes is 1 Strictly speaking what we have defined above is a restricted set of probability spaces known as discrete spaces this means that the set of sample points is either finite or countably infinite such



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