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Probability Theory Part 1: Basic Concepts

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Sample Space - Events Sample Point The outcome of a random experiment Sample Space S The set of all possible outcomes Discrete and Continuous Events A set of outcomes, thus a subset of S Certain, Impossible and Elementary

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Set Operations Union Intersection Complement Properties Commutation Associativity Distribution De Morgan’s Rule S

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Axioms and Corollaries Axioms If If A 1, A 2, … are pairwise exclusive Corollaries

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Computing Probabilities Using Counting Methods Sampling With Replacement and Ordering Sampling Without Replacement and With Ordering Permutations of n Distinct Objects Sampling Without Replacement and Ordering Sampling With Replacement and Without Ordering

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Conditional Probability Conditional Probability of event A given that event B has occurred If B 1, B 2,…,B n a partition of S, then (Law of Total Probability) S B1B1 B3B3 B2B2 A

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Bayes’ Rule If B 1, …, B n a partition of S then 01 1-pp 1010 1-εε ε input output Example Which input is more probable if the output is 1? A priori, both input symbols are equally likely.

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Event Independence Events A and B are independent if If two events have non- zero probability and are mutually exclusive, then they cannot be independent C AB ½ ½ ½ ½ ½ 11 1 1 11

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Sequential Experiments Sequences of Independent Experiments E 1, E 2, …, E j experiments A 1, A 2, …, A j respective events Independent if Bernoulli Trials Test whether an event A occurs (success – failure) What is the probability of k successes in n independent repetitions of a Bernoulli trial? Transmission over a channel with ε = 10 -3 and with 3-bit majority vote

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Basic Probability. Introduction Our formal study of probability will base on Set theory Axiomatic approach (base for all our further studies of probability)

Basic Probability. Introduction Our formal study of probability will base on Set theory Axiomatic approach (base for all our further studies of probability)

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