StATS: What is a binomial distribution?

A special distribution that arises in many applications where you are counting events. A similar distribution for counting events is the Poisson distribution. The binomial distribution arises from the following assumptions:

1. There are n trials.
2. Each trial has two possible outcomes, “success” or “failure”.
3. The probability of success, p, is the same for each trial.
4. Each trial is independent.

If X is the number of successes out of n trials, then X has a binomial(n,p) distribution. Here are some examples of binomial distributions:

• You select four patients for liver transplants and count the number of patients who survive for one year beyond the operation. Each patient is independent of the other patients and the probability of survival is 0.3 for each patient. Under these four assumptions, X=number of surviving patients is binomial(4,0.3).
• A couple has three children. Each birth is independent of the other births and the probability of a girl is 0.5 at each birth. Under these assumptions, X=number of girls is binomial(3,0.5).
• Twenty healthy volunteers are given a flu vaccine to see how many develop resistance to the flu virus. The volunteers are independent of one another and the probability of developing resistance is 0.94 for each volunteer. Under these assumptions, X=number of volunteers who develop resistance is binomial(20,0.94).

This work is licensed under a Creative Commons Attribution 3.0 United States License. It was written by Steve Simon on 2002-10-11, edited by Steve Simon, and was last modified on 2010-04-01. This page needs minor revisions. Category: Definitions, Category: Probability concepts.