lec1 - CS648 Randomized Algorithms Lecture notes : 1 1...

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Unformatted text preview: CS648 Randomized Algorithms Lecture notes : 1 1 Elementary Probability Theory Happening of unexpected events around us is not uncommon. It therefore makes sense to abandon the idea of a world with certainties and accept a world where we associate likelihood to each event. Probability is a quantitative measure of likelihood of an event. We are all familiar with these phrases and we have also fair amount of understanding about them : • “Probability of getting a head in a toss of a fair coin is 1/2”. • “Probability of a human getting infected by HIV during blood transfer is 0.000000001” • “Probability that six appears in two consecutive throws of a fair dice is 1/36”. For each such random experiment, there is a well defined set of all possible events (outcomes) which the experiment can results in. It is also quite natural to say that probability of any event has to be non-negative. Based on these understandings, we define the concept of a probability space which formalizes the notion of probability. Definition 1.1 Probability space ( Ω , Pr ) A finite probability space is a pair (Ω , Pr ) , where Ω is a finite set called sample space, and Pr is a real-valued function on the elements of Ω with the following rules. 1. Pr ( ω ) ≥ , ∀ ω ∈ Ω 2. ∑ ω ∈ Ω Pr ( ω ) = 1 . The elements of the sample space Ω are called the sample points or elementary events. Though each experiment will produce exactly one elementary event, we are sometimes interested in some event which is essentially a collection of elementary events. For example, if we toss a coin size times, there will be 2 6 elementary events, and an event can be defined as “at least 3 heads appear”. Definition 1.2 ( Event ) Given a probability space (Ω , Pr ) , an event is a subset of set Σ . It is quite natural to define probability of an event A as Pr ( A ) = X w ∈ A Pr ( w ) In many cases, each elementary event is equally likely. In such cases, it is quite easy to compute the prob- ability of an event : If there are k elementary events defining the space Σ and each is equally likely, then Pr ( w ) = 1 /k ; probability of an event E can thus be calculated by multiplying it with the number of ele- mentary events belonging to E . A few Examples : 1 1. Throw of two fair coins : The sample space is { HH, HT, T T, T H } and probability of each elemen- tary event is 1 / 4 . The events of interest could be : “at least one head appears in the outcome”, “head appears both times”, “two outputs are different”. 2. Throw of two dice : The sample space is { ( i, j ) | 1 ≤ i, j ≤ 6 } . and probability of each elemen- tary event is 1/6. A few events of interest are : “the sum of the numbers appearing on top faces is even”, “the numbers appearing on the top faces are different”, “the numbers appearing on the top face differ at least by 2”....
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This note was uploaded on 11/24/2009 for the course CS CS648 taught by Professor Surenderbaswana during the Spring '08 term at University of Massachusetts Boston.

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lec1 - CS648 Randomized Algorithms Lecture notes : 1 1...

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