Definition

An experiment is a binomial experiment if it satisfies the four requirements below:

1. There are a fixed number of trials.

2. The trials are independent. (i.e. The outcome of any individual trial does not affect the probability of the other trials.)

3. The outcomes of each trial can be classified into two categories which are typically called success and failure.

4. The probabilities remain constant for each trial.

Example 1

Is the following experiment binomial? Roll a fair die 10 times and record the outcomes.

Solution .

Since there a six possible outcomes in each trial, the experiment is not binomial.

Example 2

Suppose you want to know the probability of rolling a three 4 times in 10 rolls of a fair die. Is this a binomial experiment?

Solution .

The fixed number of trials is n = 10. The outcome of each trial is either success, rolling a three, or failure, not rolling a three. The trials are independent. The probabilities remain constant for each trial since the probability of success (rolling a three) is 1/6 on each roll of the die and the probability of failure (not rolling a three) is 5/6 on each roll of the die. Since the experiment meets the four conditions in the definition, the experiment is a binomial experiment.

Notation Used with Binomial Experiments

n = the number of trials

p = probability of success on each trial

q = 1 - p = probability of failure on each trial

x = the number of successes in n trials