Bernoulli Trials

Many experiments of everyday and scientific interest have only two possible results: a coin shows heads or tails, a manufactured item is defective or not, a patient responds to a treatment or does not, an answer on a quiz is right or wrong. Any single performance of such a two-outcome experiment is called a Bernoulli trial, named after the mathematician Jakob Bernoulli, who first studied this kind of repeated trial around 1700.

By convention, one of the two outcomes is labelled success (S) and the other failure (F) — these are just labels and do not need to mean anything good or bad; "success" might just as well be "the item is defective" if that is the event being counted. If $p$ denotes the probability of success on a single trial and $q$ the probability of failure, then because these two outcomes are the only possibilities,

$p + q = 1$.

A sequence of Bernoulli trials becomes especially easy to analyse when it satisfies four conditions:

1. The number of trials, $n$, is fixed and finite.
2. Each trial results in exactly one of two mutually exclusive outcomes — success or failure.
3. The trials are independent of one another (the outcome of one trial has no bearing on any other).
4. The probability of success, $p$, stays the same on every single trial.

Whenever an experiment meets all four conditions, we call it a sequence of independent, identical Bernoulli trials, and this is precisely the setting in which the Binomial distribution — the subject of the next section — applies. …