The Bernoulli distribution models a single success/failure trial, and the binomial distribution counts successes across n independent trials with constant probability.

What this dot point is asking

A Bernoulli trial is a single experiment with exactly two outcomes, labelled success and failure. The binomial distribution arises when you repeat such a trial a fixed number of times and count how many successes occur.

The four binomial conditions

You may only use the binomial model when all four hold:

1. A fixed number $n$ of trials.
2. Each trial has only two outcomes (success/failure).
3. Trials are independent.
4. The probability of success $p$ is constant across trials.

The binomial probability formula

If $X\sim B(n,p)$, the probability of exactly $x$ successes is:

The three pieces are: $\binom{n}{x}$ (how many ways to choose which trials succeed), $p^x$ (those successes), and $(1-p)^{n-x}$ (the remaining failures).

"At least" and "at most" probabilities

For cumulative questions, add the relevant terms - and use the complement when it is shorter.

Mean and variance

The binomial mean and variance follow directly from the parameters:

For the quiz, $E(X)=8\times 0.25=2$ correct on average, and $\operatorname{Var}(X)=8\times 0.25\times 0.75=1.5$.

Common errors

Why it matters

The binomial distribution is the most heavily examined discrete model in Stage 2 and is the bridge to the normal distribution in Topic 5, where a binomial with large $n$ is approximated by a normal curve. Recognising the four conditions also trains the modelling judgement assessed in the Mathematical Investigation.