Binomial Experiments

Criteria for a Binomial Probability Experiment

A binomial experiment is an experiment which satisfies these four conditions:

• A fixed number of trials
• Each trial is independent of the others
• There are only two outcomes
• The probability of each outcome remains constant from trial to trial.

In short: An experiment with a fixed number of independent trials, each of which can only have two possible outcomes.

(Since the trials are independent, the probability remains constant.)

The Binomial Probability Distribution

A binomial experiment is one that possesses the following properties:

1. The number of successes X in n trials of a binomial experiment is called a binomial random variable.The experiment consists of n repeated trials;
2. Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);
3. The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.

If an experiment is a binomial experiment, then the random variable X = the number of successes is called a binomial random variable.

Let's look at an examples to check your understanding.

Example 1

Consider the experiment where three marbles are drawn without replacement from a bag containing 20 red and 40 blue marbles, and the number of red marbles drawn is recorded. Is this a binomial experiment?

No! The key here is the lack of independence - since the marbles are drawn without  replacement, the marble drawn on the first will affect the probability of later marbles.