Definition
The Binomial Distribution describes the behavior of a random variable(count variable) X under the following conditions:
1. the number of trials(n) is fixed
2. trials have only 2 possible outcomes (success/failure)
3. each trial is independent of the other
4. probability of success(p) is constant throughout
X(the random variable) is a measure of the number of successes in n trials.
Example
A simple example is choosing 1 ball from a bag of 10 identical balls, each numbered (110). Once noted, the ball is returned to the bag.
A single ball is chosen on 3 separate occasions.
Success is in obtaining a '5' ball.
So the random variable X has values 0, 1, 2, 3
in other words, from our 3 tries we could have obtained:
0 fives, 1 five, 2 fives, 3 fives
On the first try, the probability of obtaining a 5 is 1/10 .
The probability of not getting a 5 is 9/10 .
Every time we dip into the bag of 10 balls, the probability of obtaining a '5' is 1/10. The probability is constant.
Getting a '5' or not getting a '5' means that there are only 2 outcomes.
Every try is independent. Previous tries do not affect the result, since previously chosen balls are returned to the bag. Every try is taken from a bag of 10 balls.
Notation B(n, p)
The full notation describing a Binomial distribution is:
where,
X a random variable (0, 1, 2, 3,...)
~ B 'is distributed Binomially'
n number of trials
p probability of single trial 'success'
Example (continued from above)
Say that there are only 3 tries of attempting to take a 5ball from a bag of 10 balls.
So n = 3 and p = 1/10.
The possible number of 5's taken in the 3 trials is summarized by the values of the random variable X .
X = 0, 1, 2, 3
using the Binomial notation,
Limits
The population size(n) of a Binomial Distribution must be much larger than the sample size(r).
The distribution only applies to trials from a simple random sample, where n is at least x10 times > r .
Outside this limit, results do not follow the equation.

