The binomial distribution

What the exam asks

Across multiple-choice and free-response questions, you will be asked to:

• Identify or justify whether a random variable follows a binomial distribution by checking all four required conditions in context.
• Calculate and interpret the mean $\mu_X$ and standard deviation $\sigma_X$ of a binomial random variable.
• Calculate exact probabilities $P(X = x)$ and cumulative probabilities such as $P(X \le x)$ or $P(X \ge x)$ using the binomial formula or your graphing calculator, while properly communicating your work.
• Estimate binomial probabilities from a dotplot or table of simulated trials.

Core ideas

Conditions for a binomial setting (BINS)

A discrete random variable $X$ follows a binomial distribution if it counts the number of successes in a setting that satisfies four conditions, easily remembered as BINS:

Binomial probabilities

If $X$ is a binomial random variable with parameters $n$ and $p$, the probability of getting exactly $x$ successes in $n$ trials is given on the AP formula sheet by:

where $\binom{n}{x} = \frac{n!}{x!(n-x)!}$ counts the number of ways to arrange successes among trials.