Binomial Distribution

AP Statistics Lectures

by Arnold Kling

Many random processes can be mapped into two outcomes. For example:

win or lose
success or failure
heads or tails

These either-or situations are called binary outcomes.

Suppose I am at a party, and I ask a girl to dance. There are two outcomes. She can agree to dance with me, or she can turn me down.

At the party, I can ask any number of girls to dance with me. Each girl has the same two choices: agree to dance with me, or turn me down.

Assume that each girl's decision is independent of one another, and that my probability of success with each girl is the same. These are the iid assumptions that we described at the end of the last lecture.

Let n be the number of girls I ask to dance. Let X be the number of girls who agree to dance with me. We say that X is a binomial random variable, because it is the sum of binary outcomes.

Suppose that when I ask a girl to dance with me, the probability that she will agree to do so is .15. We call this p, the probability of success.

The distribution of the random variable X--my overall number of dance partners--will depend on the number of girls I ask, n, and my probability of success, p.

If I ask one girl to dance (n=1), what is the mean of X? What is the variance of X?

Start with this simple table:

Result	Value of X	Probability
She accepts	1	.15
She turns me down	0	.85

E(X) = 1(.15) + 0 (.85) = .15
E(X-X)² = (.85)²(.15) + (.15)²(.85) = (.15)(.85)(.85 + .15)
= (.15)(.85)

In general, if n=1 in a binomial process, then m_X = p and s²_X = p(1-p). Now, we can use the iid equations to find the values for m_X and s²_X for values of n>1. The formulas for the mean, variance, and standard deviation of a general binomial process are:

m_X = np
s²_X = np(1-p)
s_X = square root of [np(1-p)]