# Probability Theory - PowerPoint PPT Presentation

1 / 9

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Probability Theory

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

• Union

• Intersection

• Complement

• Properties

• Commutation

• Associativity

• Distribution

• De Morgan’s Rule

S

Axioms

If

If A1, A2, … are pairwise exclusive

Corollaries

### 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 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

• 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

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

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