Binomial Distribution
 

Definition :

In statistics the so-called binomial distribution describes the possible number of times that a particular event will occur in a sequence of observations. The event is coded binary, it may or may not occur.

Formula :

The binomial distribution (the term first used by Yule, 1911) is mathematically defined as:

f(x) = [ n!/( x! x (n-x)! ) ] x px x qn-x
for x = 0, 1, 2, ..., n

where
p     is the probability of success at each trial
q     is equal to 1-p
n     is the number of independent trials

Illustrations :

A classic example that is used often to illustrate concepts of probability theory, is the tossing of a coin. If a coin is tossed 4 times, then we may obtain 0, 1, 2, 3, or 4 heads. We may also obtain 4, 3, 2, 1, or 0 tails, but these outcomes are equivalent to 0, 1, 2, 3, or 4 heads. The likelihood of obtaining 0, 1, 2, 3, or 4 heads is, respectively, 1/16, 4/16, 6/16, 4/16, and 1/16.
A sample calculation
In the figure on this page the distribution is shown with p = 1/2 Thus, in the example discussed here, one is likely to obtain 2 heads in 4 tosses, since this outcome has the highest probability. 
 

Consider another example. Say you are driving down on Beaver Avenue. There are three traffic lights that you have to go past to get to your friend's apartment. Assume that each traffic light is green for 20 seconds and red for 40 seconds. What is the probability that you will have to stop at none, one, two or all three lights ?
Let us first calculate the probability of the light being green.
p = 20 / ( 20 + 40 ) = 1/3
The probability of the light not being green is
q = 40 / ( 20 + 40 ) = 2/3 or can be also calculated as q = 1 - p = 2/3
Now let us find the probability that you will not encounter any red light i.e. x = 0

f(0) = [ 3!/( 0! x (3-0)! ) ] x (1/3)0 x (2/3)3-0

        = [ 6/( 1 x 6 ) ] x (1/3)0 x (2/3)3-0

        = (2/3)3

        = 8/27

Similarly the probabilties that you might have to stop at 1,2 or all three lights are  2/9, 2/9 and 8/27

   Application :

The binomial distribution is used when a researcher is interested in the occurrence of an event, not in its magnitude. For instance, in a clinical trial, a patient may survive or die. The researcher studies the number of survivors, and not how long the patient survives after treatment. Another example is whether a person is ambitious or not. Here, the binomial distribution describes the number of ambitious persons, and not how ambitious they are.
The binomial distribution is specified by the number of observations, n, and the probability of occurence, which is denoted by p.

Other situations in which binomial distributions arise are quality control, public opinion surveys, medical research, and insurance problems.
 

Applet :

For an illustration of the binomial distribution go to http://www-stat.stanford.edu/~naras/jsm/example5.html

Excel Function :

These functions can be accessed by clicking on Insert and then choosing Function from the drop down menu.
The Excel function for finding a binomial distribution for a given data set is :
Binomdist(number_s, trials, probability_s, cumulative)
This returns the individual term binomial distribution probability

Further comments :

Poisson Limit

If the probability p is small and the number of observations is large the binomial probabilities are hard to calculate. In this instance it is much easier to approximate the binomial probabilities by poisson probabilities. The binomial distribution approaches the poisson distribution for large n and small p. In the movie we increase the number of observations from 6 to 50, where the parameter p in the binomial distribution remains 1/10. The movie shows that the degree of approximations improves as the number of observations increases.

For moderate values of p, the binomial distribution approaches the normal distribution if the number of observations is large. This is an example of the central limit theorem.

The above text was taken from http://www.stattucino.com/berrie/binomial.html