The conditions for a binomial model, the binomial probability formula, calculating individual and cumulative probabilities, the mean of a binomial distribution, and using the model in context.

What this dot point is asking

AQA wants you to recognise when a binomial model applies, use the binomial probability formula, calculate individual and cumulative probabilities (using the calculator's binomial functions on Paper 3), find the mean of a binomial distribution, and apply the model to real contexts while stating its assumptions. The binomial is also the model behind one-tailed and two-tailed hypothesis tests for a proportion, so it links directly forward.

Conditions for a binomial model

A classic failure is sampling without replacement from a small population: the probability of success changes after each draw, breaking both the constant-probability and independence conditions. Sampling from a very large population, by contrast, keeps $p$ effectively constant, so the binomial is a reasonable model.

The probability formula

The three factors each have a meaning: $\binom{n}{r}$ counts the orderings, $p^r$ is the probability of the $r$ successes, and $(1-p)^{n-r}$ is the probability of the $n - r$ failures. Understanding this structure helps you avoid dropping the binomial coefficient.

Cumulative probabilities

For ranges such as $P(X \le r)$, $P(X \ge r)$ or $P(a \le X \le b)$, use the cumulative binomial function. Two conversions are essential: $P(X \ge r) = 1 - P(X \le r - 1)$, and $P(X > r) = 1 - P(X \le r)$. Read inequalities carefully, since "fewer than $3$" means $P(X \le 2)$ while "at most $3$" means $P(X \le 3)$.

Mean and variance

The mean gives a quick sanity check: if you compute a probability for a value of $r$ far from $np$ and it comes out large, you have likely made an error.

Modelling in context

Real questions rarely say "binomial"; you have to recognise it. Look for a fixed number of repeated trials, a clear success or failure for each, a constant success probability, and independence. Phrases such as "$8$ percent of components are faulty" or "each seed germinates with probability $0.7$" give you $p$, and "a sample of $20$" gives $n$. Define your variable explicitly, for example "let $X$ be the number of faulty components, $X \sim B(20, 0.08)$", because the examiner rewards a clear statement of the model.

When you are asked to criticise a binomial model, the usual targets are the independence and constant-probability assumptions. Components made on the same machine may fail together if the machine drifts, breaking independence; sampling without replacement from a small batch changes $p$ on each draw. Stating which specific assumption is doubtful, and why, earns the evaluation marks that a generic "it might not be accurate" does not.