The Binomial Distribution

Perhaps the most commonly used discrete probability distribution is the binomial distribution. An experiment which follows a binomial distribution will satisfy the following requirements (think of repeatedly flipping a coin as you read these):

1. The experiment consists of n identical trials, where n is fixed in advance.
2. Each trial has two possible outcomes, S or F, which we denote ``success'' and ``failure'' and code as 1 and 0, respectively.
3. The trials are independent, so the outcome of one trial has no effect on the outcome of another.
4. The probability of success, , is constant from one trial to another.

The random variable X of a binomial distribution counts the number of successes in n trials. The probability that X is a certain value x is given by the formula

where Recall that the quantity , ``n choose x,'' above is

where

We could use the formulas previously given to compute the mean and variance of X. However, for the binomial distribution these will always be equal to

NOTE:\ A random variable X which follows the binomial distribution can be abbreviated , where is read ``is distributed as.''

NOTE:\ A particularly important example of the use of the binomial distribution is when sampling with replacement (this implies that is constant).

EXAMPLE:\ Suppose we have 10 balls in a bowl, 3 of the balls are red and 7 of them are blue. Define success S as drawing a red ball. If we sample with replacement, P(S)=0.3 for every trial. Let's say n=20, then and we can figure out any probability we want. For example,

The mean and variance are