Probability Theory and its Applications

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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