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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

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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