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Foundations of Probability: Part 1 - Basic Concepts and Computation Methods

This guide covers fundamental concepts in probability theory, including sample space, events, axioms, corollaries, computing probabilities with counting methods, conditional probability, Bayes' Rule, event independence, Bernoulli trials, and sequences of independent experiments. Learn about sample points, discrete and continuous events, set operations, and more. Explore practical examples like computing probabilities for different scenarios and understanding event independence in sequences of independent experiments.

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Foundations of Probability: Part 1 - Basic Concepts and Computation Methods

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  1. Probability Theory Part 1: Basic Concepts

  2. 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

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

  4. Axioms If If A1, A2, … are pairwise exclusive Corollaries Axioms and Corollaries

  5. 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

  6. Conditional Probability • Conditional Probability of event A given that event B has occurred • If B1, B2,…,Bn a partition of S, then (Law of Total Probability) S B1 B2 A B3

  7. Bayes’ Rule • If B1, …, Bn a partition of S then Example Which input is more probable if the output is 1? A priori, both input symbols are equally likely. input 0 1 1-p p output 0 1 0 1 1-ε ε ε 1-ε

  8. Event Independence A B • Events A and B are independentif • If two events have non-zero probability and are mutually exclusive, then they cannot be independent 1 1 ½ ½ C 1 ½ 1 1 ½ ½ 1

  9. Sequences of Independent Experiments E1, E2, …, Ej experiments A1, A2, …, Aj 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 Sequential Experiments

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