A Concrete Introduction to Higher Algebra, 2nd Edition

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Columbia University Handout 3 E6291: Topics in Cryptography February 2, 1999 Professor Luca Trevisan Notes on Discrete Probability The following notes cover, mostly without proofs, the basic notions and results of discrete probability. Two important topics (the birthday paradox and universal hash functions ) are not covered, and they will be the subject of future notes. 1 Basic Definitions In cryptography we typically want to prove that an adversary that tries to break a certain protocol has only minuscule (technically, we say “negligible”) probability of succeeding. In order to prove such results, we need some formalism to talk about the probability that certain events happen, and also some techniques to make computations about such proba- bilities. In order to model a probabilistic system we are interested in, we have to define a sample space and a probability distribution . The sample space is the set of all possible elementary events , i.e. things that can happen. A probability distribution is a function that assigns a non-negative number to each elementary event, this number being the probability that the event happen. We want probabilities to sum to 1. Formally, Definition 1.1 For a finite sample space Ω and a function Pr : Ω R , we say that Pr is a probability distribution if 1. Pr ( a ) 0 for every a Ω ; 2. a Ω Pr ( a ) = 1 . For example, if we want to model a sequence of three coin flips, our sample space will be { Head, Tail } 3 (or, equivalently, { 0 , 1 } 3 ) and the probability distribution will assign 1 / 8 to each element of the sample space (since each outcome is equally likely). If we model an algorithm that first chooses at random a number in the range 1 , . . . , 10 200 and then does some computation, our sample space will be the set { 1 , 2 , . . . , 10 200 } , and each element of the sample space will have probability 1 / 10 200 . We will always restrict ourselves to finite sample spaces, so we will not remark it each time. Discrete probability is the restriction of probability theory to finite sample spaces. Things are much more complicated when the sample space can be infinite. An event is a subset A Ω of the sample space. The probability of an event is defined in the intuitive way
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2 Handout 3: Notes on Discrete Probability Pr [ A ] = a A Pr ( a ) (Conventially, we set Pr [ ] = 0.) We use square brackets to remind us that now we are considering a different function: while Pr ( · ) is a function whose inputs are elements of the sample space, Pr [ · ] is a function whose inputs are subsets of the sample space. For example, suppose that we want to ask what is the probability that, when flipping three coins, we get two heads. Then Ω = { 0 , 1 } 3 , Pr ( a ) = 1 / 8 for every a Ω, we define A as the subset of { 0 , 1 } 3 containing strings with exactly two 1s, and we ask what is Pr [ A ]. As it turns out, A has 3 elements, that is 011 , 101 , 110, and so Pr [ A ] = 3 / 8. Very often, as in this example, computing the probability of an event reduces to counting the number of elements of a set.
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  • Spring '02
  • Trevisan
  • Probability, Probability theory

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