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 $p$ of a success is the same for any trial (so the probability $q = 1 − p$ 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, $n$.
2. There are only two possible outcomes, called “success” and, “failure,” for each trial. The letter $p$ denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial.
3. The $n$ trials are independent and are repeated using identical conditions.

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

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