Introduction to Probability Theory and its Applications

The probability of an occurrence is expressed on a size from 0.0 to 1.0

Way it will never happen

Way that it is certain to happen

The chance of an event is the number of times that a specific event occurs relative to the sum of all possible events that can occur.

Theoretical probability

Sometimes we know from the theory of the matter what the probability of an event is, e.g. regular dice or tossing coins.

Probability of an event

The proportion of times an event occurs separated by the frequency of all other events that can occur, e.g. 246 acquittals out of 14,573 cases acquitted, fined, deferred, probated, or sent to prison (‘246/14573’ = p = 0.0169)

Relative frequency

How often an event occurs relative to all other events that occurred in the experiment

Mutually exclusive events

Two or more events which cannot ensue mutually, .e.g. acquitted and sent to prison, the probability = 0.0

Conditional probability

The probability of event A happening, given that event B has already occurred, e.g. probability of going to prison (A) given that the crook was put on probation (B).

Independent events

Two procedures A & B are considered independent if the conditional probability P(A çB) = P(A), e.g. chance of acquittal (A) given that it is raining outside (B)

Addition Rule of Probability Theory

P(A or B) = P(A) + P(B) – P(A and B)

Multiplication Rule of Probability Theory

P (B) P (A|B)

Probability Distributions

Those indicate the probability of specific events happening for a phenomenon distributed in a particular manner.

In statistics, probability distributions are used to briefly, expect, and aid in decision making.

Binomial distribution

Normal distribution

t distribution

F distribution

Chi-square distribution

Poisson distribution

Examples for Probability Theory and its Applications:

Probability Theory – Example 1:

A fair coin is flipped 3 times. Let S be the sample space of 8 possible outcomes, and let X be a random variable that assignees to an outcome the number of heads in this outcome.

Solution:

where X(S) = {0, 1, 2, 3} is the range of X, which is the number of heads, and

S={ (TTT), (TTH), (THH), (HTT), (HHT), (HHH), (THT), (HTH) }

X(TTT) = 0

X(TTH) = X(HTT) = X(THT) = 1

X(HHT) = X(THH) = X(HTH) = 2

X(HHH) = 3

The probability distribution (pdf) of random variable X is given by

P(X=3) = ‘1/8’ , P(X=2) = ‘3/8’ , P(X=1) = ‘3/8’ , P(X=0) = ‘1/8’ .

Probability Theory Example 2:

What is the probability of drawing either a Jack or a Heart from a deck of cards?

P(J or H) = P(J) + P(H) – P(J and H)

P(J or H) = P(‘4/52′ ) + P(’13/52’ ) – P(‘1/52’ )

P(J or H) = (0.0769 + 0.2453) – (0.01923)

P(J or H) = 0.3077

Probability Theory Applications:

The Probability and Its Applications sequence issues research monographs, with the expository excellence to make them useful and available to advanced students, in probability and stochastic processes, with a particular focus on:

Basics of probability containing stochastic analysis and Markov and other stochastic processes

Applications of probability in analysis

Application Point processes, random sets, and other spatial models

Branching processes and other models of population growth

Genetics and other stochastic models in Application biology

Information theory and signal processing

Communication networks

Application Stochastic models in operations research

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