Data often arise in the form of counts or proportions which are realizations of a discrete random variable. A common situation is to record how many times an event occurs in n repetitions of an experiment, i.e., for each repetition the event either occurs (a "success") or it does not (a "failure").

Binomial Experiment

More specifically, consider the following experimental process:

  1. There are n trials.
  2. Each trial results in a success or a failure.
  3. The probability of a success, p, is constant from trial to trial.
  4. The trials are independent.

An experiment satisfying these four conditions is called a binomial experiment. The outcome of this type of experiment is the number of successes, i.e., a count. The discrete variable X representing the number of successes is called a binomial random variable. The possible counts, X = 0,1,2, ..., n, and their associated probabilities define the binomial distribution, denoted by B(n,p).

Binomial Distribution

The following Binomial Applet can be used to experiment with the binomial distribution.

Binomial Moments

The binomial mean, or the expected number of successes in n trials, is E(X) = np. The standard deviation is Sqrt(npq), where q = 1-p. The standard deviation is a measure of spread and it increases with n and decreases as p approaches 0 or 1. For a given n, the standard deviation is maximized when p = 1/2.


Example #1 shows how probabilities and quantiles are computed when a student guesses on a multiple-choice test.

Example #2 compares the distributions of the number of delinquents and non-delinquents who wear glasses.


Exercise #1 computes probabilities and quantiles for the number of patients surviving a heart-transplant operation.

Exercise #2 compares the distributions of the number of patients recovering from a cold using two drugs.

Exercise #3 computes probabilities and quantiles for the number of males in a litter of geckos.