Modeling With the Binomial Distribution

Recognize when a real-world situation can be modeled by a binomial distribution, identify the parameters $n$ and $p$ , and compute probabilities of single and cumulative outcomes — including the use of the complement for 'at least' events.

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Tutorial

When Does a Situation Follow a Binomial Distribution?

A random variable $X$ follows a binomial distribution with parameters $n$ and $p$ , written $X \sim B(n, p)$ , when the following four conditions hold:

There is a fixed number of trials, $n$ .
Each trial has only two possible outcomes, labeled success and failure.
The probability of success, $p$ , is the same on every trial.
The trials are independent.

To model a real-world situation with the binomial distribution, we:

Decide what counts as a success on a single trial.
Identify $p$ , the probability of success on a single trial.
Identify $n$ , the total number of trials.
Let $X$ be the number of successes; then $X \sim B(n, p)$ .

For example, suppose we flip a fair coin $10$ times and let $X$ be the number of heads. Calling 'heads' a success gives $p = 0.5$ and $n = 10$ , so

X \sim B(10,\, 0.5).