Discrete Distributions


The Binomial Distribution

[Yang Hui triangle]Random variables with a Binomial distribution are generated from a process that follows the following five rules:

  • The number of trials, n, is known.
  • Each trial has two possible outcomes. We will refer to these as either “Success” and “Failure,” or a 1 and 0, respectively.
  • The probability of success in each trial, p, does not change from trial to trial.
  • The trials are independent; that is, the outcome of each trial is not influenced by the outcome of any other trial.
  • The random variable is the number of successes in those n trials.

In short, the Binomial random variable is the sum of n independent and identically distributed Bernoulli random variables.


The Binomial distribution has two parameters. The first is the number of trials, n. The second parameter is the success probability, p. This latter parameter may also be represented as π, depending on the source. Note that the sample space is defined as all integers between 0 and n, inclusive.


Please select the aspect of the Binomial distribution you would like to work with:

Binomial Example

Here is a graphic of a sample Binomial distribution:

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 1 2 3 4 5

In this example,

X ~ Bin(n=5, p=0.61)

The sample space is listed across the bottom, S = {0, 1, 2, … 5}. The height of each bar represents the probability of that elementary event.

© Ole J. Forsberg, Ph.D. 2017. All rights reserved.   .