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Probability Theory

The word "probability" is one of of those unique ones that you come across and you "feel" as if you know exactly what it represents. This may be somewhat misleading. First of all it does not mean "probably". Secondly, many mathematicians and logicians have spent a lot of time defining a very precise science called "probability" or "probability theory".

Definitions:

Definition #1 ( Relative frequency definition)

Prob(A) = ( number of outcomes in which A occurs )
( total number of times trial was repeated )

Definition #2 ( Classical Approach )

Prob(A) = ( number of ways A can occur ) = s
( number of different simple events)

These definitions can only be brought out by concrete examples.

Examples we will deal with often use a deck of 52 playing cards because the events are so clearly defined.

At the University of Ottawa, David MacDonald is a professor in mathematics and does call himself a probablist. One of his areas of interest is working on switching simulations associated with fibre-optic information transportation. Switches are often the expensive components of the information highway so it is useful to perform optimization calculations.

Probability theory is needed with information packet transfers since the new communication lines are no longer deterministic.

Correct answer:

Probability theory starts with basic definitions:

An experiment is any process that allows researchers to obtain observations or data points.	An event is any collection of results or outcomes of an experiment
A simple event is an outcome that cannot be further broken down into simpler components	The sample space for a procedure consists of all possible simple events

Events are important and are represented by letters
such as A , B , and C.

The probability of an event A is given as Prob(A) or in some
textbooks as P(A).

Prob(A) = probability of event A
Prob(A or B) = probability of event A or event B.

We will see easier uses of probability theory as well. One obvious area of probability theory is reading suitable and factual past information. John wants take his wife and family to Vancouver for a long-needed vacation.
Since he has lots of time he has researched the safest ways for his family to go to Vancouver and return:

Question: John and family can go to Vancouver by

a) rental car
b) light airplane
c) commercial jet
d) train

Which transportation is most dangerous?

Which mode of transportation is safest? Explain.

The science of probability theory

This science of probability theory is based on axioms similar to the foundations of all good mathematical areas. A first course in statistics does not require a complete review of these assumptions but we will study a lot of useful fundamentals.

The theory and laws of probability should apply to everyday events and situations. The most common everyday application is that of the weather report. If we know that there is an 80% probability of 15 centimeters of snow tomorrow than we may prepare ourselves accordingly.