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Probability Theory and Concepts

What are the odds -

What is Probability?

A probability is a measure of the likelihood that an event in the future will happen.  We assume a probability between 0 and 1 (or Zero Percent and 100 Percent).

  • A probability is expressed in terms of a value between zero and one. The values describe the relative likelihood an event will occur.  The closer the probability is to 0, the lesser the chance it will occur.  The closer the probability is to 1, the higher the chances it will occur.

Basic relationships of probability: According to the text, the terms probability, chance, and likelihood are often used interchangeably (Lind, 2002).  

Information provided by: Probability Theory , Lauri Gilliam , Laura Grider , Sharol Thomsen ; University of Phoenix , QNT 530 , Statistics and Research Methods for Managerial Decisions


Craig A. Stevens Explains Continuous vs Discrete Data

Two types of probability are objective and subjective probability. 

  1. Objective probability is based on observation. For example, if the farmerís almanac has rainfall data for April for the past 25 years, one might use that data to complete an analysis and say that it rains 1 out of 5 days in April.
  2. Subjective probability, on the other hand, has little or no historical data to support it.  Calculating subjective probability requires making an estimate based on available opinions and any other information.  Subjective probability is based on whatever information is available.  

OBJECTIVE PROBABILITY:  Classical vs. Empirical 

Objective probability can be broken down into a) classical probability and b) empirical probability.
  1. Classical probability is based on the assumption that any outcomes of the experiment are equally likely.  There is no past history used to predict a future occurrence.  The classical definition applies when there are n equally likely outcomes. Calculating classical probability is quite simple:

Classical probability goes on to consider that events may be mutually exclusive or collectively exhaustive.  

  • Mutually exclusive events mean that only one event can occur at a time.  No others can occur simultaneously.  

  • Collectively exhaustive events require that at least one of the outcomes will occur every time the experiment is conducted.

  1. Empirical Probability is based on using the frequency of an event happening in the past to predict when it may happen in the future.  The empirical definition applies when the number of times the event happens is divided by the number of observations.  The formula for empirical probability is:



Classical vs. Empirical Differ:

These two types of probability differ in that classical probability requires no experiment to be determined.  It is a logical calculation based on the theory that all outcomes are equally as likely to occur.  Empirical calculations must have historical occurrence data in order to project the future likelihood of an event happening.  If the situation begins with a statement regarding past data, it would require empirical calculation.  If a situation begins with a statement that all factors are equally weighted, it would be better suited to classical calculation.

Three important concepts:

  1. Experiment = the observation of some activity or the act of taking some measurement.
  2. Outcome = the particular result of an experiment.
  3. Event = the collection of one or more outcomes of an experiment.
  • Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time.
  • Events are independent if the occurrence of one event does not affect the occurrence of another.
  • Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.




Examples of subjective probability give in the book are:

  1. estimating the probability the Washington Redskins will win the Super Bowl this year.
  2. estimating the probability mortgage rates for home loans will top 8 percent.

Probability Distribution: 

Found by CAROL DOUGLAS (UoP 2005)

An interesting web page that describes discrete random variables and probability distributions, how to use cumulative probabilities, expected values of discrete random variables and continuous random variables.   It is specific to biological services but noticed the examples it gave where similar to our text (coin toss). 

Found by Charriet Womble (UoP 2005)

Rules Related to Probability of Events

1).  Special Rule of Addition: the formula for mutually exclusive events is:

                                       P(A or B) = P(A) + P(B)  

 If two events A and B are mutually exclusive, the special rule of addition states that the probability of A or B occurring equals the sum of their respective probabilities.

2). Complement Rule: The Complement Rule is used to determine the probability of an event occurring by subtracting the probability of the same event NOT occurring from 1.  For example, Bill Hall says there is a 40% chance of rain today.  Using the complement rule, we would subtract the chance of it NOT raining today (60%) from 1.

1 - .60 = .40     (Billís forecast).

If P(A) is the probability of event A and P(~A) is the complement of A, then

 P(A) + P(~A) = 1 


P(A) = 1 - P(~A).

Here is a Venn diagram illustrating the complement rule:

3).  The General Rule of Addition:

The Special Rule of Addition illustrated above will not always work because the outcomes must be mutually exclusive.  If one event occurs, none of the others can occur simultaneously.  Therefore, it is possible to isolate the probability of one specific event or combination of events from the probability that this event will NOT occur.

If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: 

P(A or B) = P(A) + P(B) - P(A and B)

Here is a Venn diagram illustrating the general rule of addition:

4).  Joint Probability

A joint probability measures the likelihood that two or more events will happen concurrently. An example given in our class (based on text) was the event that a student has both a stereo and TV in his or her dorm room.

Properties of joint probability links

found by RACHEL PARROTT (UoPhx QNT 554, 2005)

5).  The Special Rule of Multiplication

The special rule of multiplication requires that two events A and B are independent.  Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. Then: 

P(A and B) = P(A)P(B)

6).  Conditional Probability

A conditional probability is the probability of a particular event occurring, given that another event has occurred. The probability of the event A given that the event B has occurred is written 


Conditional probability applies when one is attempting to predict the probability of a specific event, given that another event has occurred.  For example, if you are running 20 minutes late for work, what are the odds that you will get stopped by the police for speeding?

P(A and B) = P(A)P(B|A)

7).  General Rule of Multiplication

The general rule of multiplication is used to find the joint probability that two events will occur.  For two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred.

P(A and B) = P(A)P(B/A) 


P(A and B) = P(B)P(A/B)

8).  Decision Tree Diagrams

A tree diagram is useful for portraying conditional, joint probabilities, and for analyzing business decisions involving several stages.  Uses statically values to show the different route one can take.

9).  Bayesí Theorem

Is a method for revising a probability given additional information.

"Bayesian inference has applications in artificial intelligence and expert systems. Bayesian inference techniques have been a fundamental part of computerized pattern recognition techniques since the late 1950s. There is also an ever growing connection between Bayesian methods and simulation Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while the graphical model structure inherent to all statistical models, even the most complex ones, allows for efficient simulation algorithms like the Gibbs sampling and other Metropolis-Hastings algorithm schemes.

As a particular application of statistical classification, Bayesian inference has been used in recent years to develop algorithms for identifying unsolicited bulk e-mail spam. Applications which make use of Bayesian inference for spam filtering include Bogofilter, SpamAssassin and Mozilla. Spam classification is treated in more detail in the article on the naive Bayes classifier.

In some applications fuzzy logic is an alternative to Bayesian inference. Fuzzy logic and Bayesian inference, however, are mathematically and semantically not compatible: You cannot, in general, understand the degree of truth in fuzzy logic as probability and vice versa."

Submitted by Cyndi Cox, (UoP 2005)


10).  The multiplication formula 

Indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.

11).  Permutations

A permutation is any arrangement of r objects selected from n possible objects and the order of arrangement is important.  Think of a combination lock...the combination lock is a dirty lie.  It should be called a permutation lock.  Because order of the numbers is important.

12).  Combinations:

A combination is the number of ways to choose r objects from a group of n objects without regard to order.  Think money...if I give you ones and twenties it does not mater which ones I give you long as I always give you the same number of ones and twenties. Think of a combination lock...the combination lock is a dirty lie.  It should be called a permutation lock.


Lind, et al. (2002).  Statistical Techniques in Business & Economics (11th ed). McGraw-Hill.



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