Binomial Distribution

Many experiments boil down to just two possible outcomes — a coin toss lands heads or tails, an answer is correct or incorrect, a component is defective or not. Whichever outcome is being counted is labelled a success, and the other a failure. Each repetition of such an experiment is called a trial.

A collection of trials is called a sequence of Bernoulli trials when it satisfies four conditions: the number of trials is finite, the trials are independent of one another, each trial has exactly two possible outcomes (success or failure), and the probability of success stays the same across every trial. This last condition is easy to lose — drawing items with replacement keeps the probability of success constant from trial to trial, but drawing without replacement changes it after every draw, so those trials do not qualify as Bernoulli trials.

Denoting the probability of success in a single trial by $p$ and the probability of failure by $q$, note that $p + q = 1$. When a fixed number $n$ of Bernoulli trials is performed, the probability of getting exactly $r$ successes is given by the binomial probability formula:

Here $n$ is the number of trials, $r$ the number of successes, $p$ the probability of success in a single trial, and $q = 1-p$ the probability of failure. Each term $P(r \text{ successes})$ is in fact the $(r+1)^{\text{th}}$ term in the binomial expansion of $(q+p)^n$ — which is exactly why this family of distributions is called the Binomial distribution. …