Key Concepts

• Binomial experiments consisted of a fixed number of independent trials.
• The probability of success and failure is the same in each trial of a binomial experiment.

Glossary

Bernoulli trials: an experiment with the following characteristics:

1. There are only two possible outcomes called “success” and “failure” for each trial.
2. The probability [latex]p[/latex] of a success is the same for any trial (so the probability [latex]q = 1 − p[/latex] of a failure is the same for any trial).

binomial experiment: a statistical experiment that satisfies the following three conditions:

1. There are a fixed number of trials, [latex]n[/latex].
2. There are only two possible outcomes, called “success” and, “failure,” for each trial. The letter [latex]p[/latex] denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial.
3. The [latex]n[/latex] trials are independent and are repeated using identical conditions.

binomial probability distribution: a discrete random variable [latex](RV)[/latex] that arises from Bernoulli trials; there are a fixed number, [latex]n[/latex], of independent trials. The notation is: [latex]X \sim B(n, p)[/latex]. The mean is [latex]\mu = np[/latex] and the standard deviation is [latex]\sigma = \sqrt{np(1-p)}[/latex]. The probability of exactly [latex]x[/latex] successes in [latex]n[/latex] trials is

[latex]P(X=x) =(_{x}^{n})p^{x}(q)^{n-x}, \mathrm{where} (_{x}^{n}) = \frac{n!}{x!(n-x)!}[/latex]