## BINOMIAL DISTRIBUTION

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:

- There are
*n* trials.
- Each trial results in a success or a failure.
- The probability of a success,
*p*, is constant from trial to trial.
- 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.

**Examples**

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.

**Exercises**

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.