Probability Theory and Concepts
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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).
Basic relationships of probability: According to the text, the terms probability, chance, and likelihood are often used interchangeably (Lind, 2002).
Two types of probability are objective and subjective probability.
OBJECTIVE PROBABILITY: Classical vs. EmpiricalObjective probability can be broken down into a) classical probability and b) empirical probability.
Classical vs. Empirical Differ:
Three important concepts:
Examples of subjective probability give in the book are:
Rules Related to Probability of Events
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
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
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.
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.
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.
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.
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 first...as 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).