OxfordAQA A-Level Mathematics: Bernoulli and Binomial Distributions (9660)

This guide covers S1.3 Bernoulli and Binomial Distributions, from Unit PSM1 (Statistics) in OxfordAQA International AS and A-level Mathematics (9660), Version 5.2, International AS exams May/June 2018 onwards, International A-level exams May/June 2019 onwards.

Syllabus coverage

OXFORDAQA INTERNATIONAL AS AND A-LEVEL MATHEMATICS (9660) — S1.3 BERNOULLI AND BINOMIAL DISTRIBUTIONS

• Conditions for application of a Bernoulli distribution
• Mean and variance of a Bernoulli distribution, including derivations of E(X) = p and Var(X) = p(1 − p)
• The binomial distribution, introduced as the sum of independent Bernoulli trials
• Calculation of probabilities using the formula and tables, including use of ⁿCₓ (n choose x) notation
• Mean, variance and standard deviation of a binomial distribution, including deductions of np and np(1 − p) from the corresponding Bernoulli values

How to approach it

Build the binomial distribution up from the Bernoulli distribution deliberately, rather than learning binomial formulas as a standalone fact set — this is exactly how the specification frames it. A single Bernoulli trial has only two outcomes (success, with probability p, and failure, with probability 1 − p), with mean p and variance p(1 − p). A binomial distribution is what results from summing n independent Bernoulli trials, so its mean (np) and variance (np(1 − p)) are direct scaled-up consequences of the single-trial values — being able to explain this derivation, not just quote the formulas, is what the specification’s own “Additional information” column signals is expected.

Before applying the binomial formula, always check the four conditions for a binomial distribution explicitly: a fixed number of trials, n; each trial has only two possible outcomes; the probability of success, p, is constant across all trials; and the trials are independent of each other. Exam questions sometimes describe a scenario that violates one of these conditions (for example, sampling without replacement from a small population, which breaks independence), and identifying that the binomial model does not strictly apply is itself an assessable skill.

This specification also expects you to be able to use tables to find binomial probabilities directly, without computing every term by hand, for standard combinations of n and p. Practise both routes — the full formula calculation and reading a cumulative binomial table — since exam questions may specify which method to use, or may reward either approach when finding a cumulative probability such as P(X ≤ 3) across several individual terms.

Worked example: calculating a binomial probability

A factory tests components with a 0.1 probability of a randomly selected component being defective. In a sample of 8 components, find the probability that exactly 2 are defective.

Step 1: confirm binomial conditions
        n = 8 (fixed number of trials)
        p = 0.1 (constant probability of "success" = defective)
        independent trials assumed

Step 2: apply the binomial probability formula
        P(X = x) = (nCx) x p^x x (1-p)^(n-x)
        P(X = 2) = (8C2) x (0.1)^2 x (0.9)^6

Step 3: calculate
        8C2 = 28
        P(X = 2) = 28 x 0.01 x 0.531441 ~= 0.149

Showing the binomial coefficient calculation as a distinct step, rather than folding it silently into a single line, is good exam practice since it allows partial credit even if the final arithmetic slips.

Key terms to define precisely

Bernoulli trial — a single random experiment with exactly two possible outcomes, conventionally labelled success (probability p) and failure (probability 1 − p). Bernoulli distribution — the probability distribution of a single Bernoulli trial, with mean E(X) = p and variance Var(X) = p(1 − p). Binomial distribution — the probability distribution of the number of successes in n independent, identically distributed Bernoulli trials, written X ~ B(n, p). Independent trials — trials whose outcomes do not affect one another, a condition required for the binomial model to apply exactly; sampling without replacement from a small, finite population is a common situation where trials are not truly independent. ⁿCₓ (n choose x) — the binomial coefficient, giving the number of distinct ways to arrange x successes among n trials, calculated as n! / (x!(n − x)!). Every one of these terms appears in the specification’s own wording for S1.3, so using them precisely — rather than paraphrasing loosely — is directly rewarded in how this content is assessed.

Common mistakes

Applying the binomial formula without first checking that the four conditions (fixed n, two outcomes, constant p, independence) genuinely hold in the scenario described. Confusing the mean and variance formulas for a single Bernoulli trial (p and p(1 − p)) with those for the full binomial distribution (np and np(1 − p)), especially under exam time pressure. Miscalculating the binomial coefficient ⁿCₓ, or omitting it from the formula entirely. Forgetting that variance, not standard deviation, is np(1 − p) — the standard deviation requires an additional square root step.

Quick revision checklist

• Learn the four conditions required for a binomial distribution to apply.
• Be able to derive np and np(1 − p) conceptually from the single-trial Bernoulli values p and p(1 − p).
• Practise calculating binomial probabilities using the full formula, including the ⁿCₓ term, not just recalling table values.
• Keep variance and standard deviation clearly distinguished — the latter requires a square root of the former.

S1.3 connects directly to S1.1 (Further probability) and S1.2 (Discrete random variables) earlier in this unit: the binomial distribution is itself a specific example of a discrete random variable, and the probability rules developed in S1.1 underpin why the binomial formula takes the form it does. Reviewing those two sub-topics alongside S1.3 makes the derivation of the binomial mean and variance considerably more intuitive than treating the formulas in isolation.

Related resources

• Bernoulli and Binomial Distributions revision notes
• Bernoulli and Binomial Distributions practice questions

Official syllabus

OxfordAQA International AS and A-level Mathematics (9660) specification, Version 5.2 — oxfordaqa.com/9660.