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# lecture17 - CS 70 Discrete Mathematics for CS Spring 2005...

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Unformatted text preview: CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 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 “experi- ment” 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 Ω , together with a probability Pr [ ϖ ] for each sample point ϖ , such that • 0 ≤ Pr [ ϖ ] ≤ 1 for all ϖ ∈ Ω . • ∑ ϖ ∈ Ω Pr [ ϖ ] = 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 as the natural numbers, or the integers, or the rationals, but not the real numbers). Later, we will talk a little about continuous sample spaces, but for now we assume everything is discrete.] CS 70, Spring 2005, Notes 17 1 Here are some examples of (discrete) probability spaces: 1. Flip a fair coin. Here Ω = { H , T } , and Pr [ H ] = Pr [ T ] = 1 2 . 2. Flip a fair coin three times. Here Ω = { ( t 1 , t 2 , t 3 ) : t i ∈ { H , T }} , where t i gives the outcome of the i th toss. Thus Ω consists of 2 3 = 8 points, each with equal probability 1 8 . More generally, if we flip the coin n times, we get a sample space of size 2 n (corresponding to all words of length n over the alphabet { H , T } ), each point having probability 1 2 n ....
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lecture17 - CS 70 Discrete Mathematics for CS Spring 2005...

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