Fundementals of Probability

 

In working with probabilities, one has to understand the basic concepts of statistics such as what qualifies as an experiment, an event, an outcome or a sample space. One has to also know the difference between events which are mutually exclusive, or exhaustive. Mutually exclusive events can be counted since they are discrete with a vertical line above them. Exhaustive events are known to continue and can not be drawn using vertical lines but can be understood with a little calculus or the help of a normal curve or normal probability table.  Some concern for probabilities is their highly objective or subjective nature.

 

First, an outcome and event may be one and the same, however, the outcome may also be the result of a given event. Probability is the liklihood of an expected event occurring. Probabilities have two basic properties. The first is that the total probability of an event is between 0 and 1. The second basic property is that the sum of all probabilities associated with one experiment is 1. The sum of all possibilities is then 100%.

 

Probability Rules

 

Sampling Distributions differ somewhat from Probability Distributions, since they involve repeated probabilities with more than one set of (discrete) data. Probability distributions involve binomial data such as with opinion polls commonly used to relate satisfaction or public opinion. These events which relate one outcome to another, over a series of repeated experiments reveal possible outcomes through a normal curve.

 

Binomials therefore refer to the outcome of success or failure.

 

Some examples of situations to be analyzed using a binomial model:

 

 

 

i) Assume that any given student has a probability 0.75 of passing a standardized test. For a sample of 20 students, calculate the probability of two or fewer failures. What are the mean and variance?

 

 

 

Let x = the number of failing students.

 

This is an example of a binomial distribution with n = 20 and pi = 0.25.

 

 

 

Note that P(failure) = 0.25 = 1 – P(pass).

 

 

 

You are interested in two or fewer failures.

 

P(2 or fewer failures) = P(x is less than or equal to 2)

 

 

 

= P(x = 2) + P(x = 1) + P(x = 0)

 

= 0.0669 + 0.0211 + 0.0032

 

= 0.0912

 

 

E(x) = n pi = 20(0.25) = 5

 

V(x) = n pi (1 – pi) = 20(0.25)(0.75) = 3.75