Probability theory
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Probability Theory. Part 1: Basic Concepts. 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. Set Operations.

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Probability Theory

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Probability theory

Probability Theory

Part 1: Basic Concepts


Sample space events

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


Set operations

Set Operations

  • Union

  • Intersection

  • Complement

  • Properties

    • Commutation

    • Associativity

    • Distribution

    • De Morgan’s Rule

S


Axioms and corollaries

Axioms

If

If A1, A2, … are pairwise exclusive

Corollaries

Axioms and Corollaries


Computing probabilities using counting methods

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


Conditional probability

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


Bayes rule

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


Event independence

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


Sequential experiments

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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