# What is a Binomial Distribution?

A Binomial Distribution is a discrete Random Sampling Distribution. The sampling is from a series of n independent trials each of which may have outcomes which fall into one of two classifications. The probability of occurrence of each classification remains constant for each trial. The outcomes of each successive trial also remains independent. This sampling situation is sometimes referred to as a Bernoulli Trial or Process.

## Calculations

The probability of exactly r occurrences in n trails may be calculated by the following Binomial formula.

Where:

- r is the number of occurrences,
- n is the sample size, and
- is the probability of occurrence.

## Other Properties

The Binomial Distribution may be approximated with a Normal Distribution with a Mean and a standard deviation as follows.

The limit of the Binomial Distribution, as n goes toward infinity, and p goes toward zero, yields a Poisson Distribution, with equal to np.

## Example

Here is an example Binomial Distribution, assuming a sample size of 10, and a probability of occurrence of 0.2. The probability of 5 or more occurrences in a sample of size 10, with a probability of occurrence of 0.2, is 0.0328.

### Binomial Distribution

p = 0.2000 n = 10 np = 2.000 Prob Equal & Equal & r at r Above Below 0 0.1074 1.0000 0.1074 |------------ 1 0.2684 0.8926 0.3758 |------------------------------- 2 0.3020 0.6242 0.6778 |----------------------------------- 3 0.2013 0.3222 0.8791 |----------------------- 4 0.0881 0.1209 0.9672 |---------- 5 0.0264 0.0328 0.9936 |--- 6 0.0055 0.0064 0.9991 |- 7 0.0008 0.0009 0.9999 | 8 0.0001 0.0001 1.0000 | 9 0.0000 0.0000 1.0000 | 10 0.0000 0.0000 1.0000 |