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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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