OxfordAQA A Level Mathematics: Bernoulli and Binomial Distributions — Revision Notes

Condensed for the final weeks. For the full explanation, use the Bernoulli and Binomial Distributions study guide.

Bernoulli trial and distribution

A Bernoulli trial has exactly two outcomes: success (probability p) and failure (probability 1 − p).

Statistic	Bernoulli	Binomial (n trials)
Mean	E(X) = p	np
Variance	Var(X) = p(1 − p)	np(1 − p)

A binomial distribution is the sum of n independent Bernoulli trials, written X ~ B(n, p). Its mean and variance are direct scaled-up versions of the single-trial values — build binomial formulas up from the Bernoulli ones rather than memorising them separately.

Four conditions for a binomial model

1. A fixed number of trials, n.
2. Each trial has only two possible outcomes.
3. The probability of success, p, is constant across trials.
4. Trials are independent of each other.

Check all four before applying the binomial formula — sampling without replacement from a small population is a common condition-breaker (it violates independence), and exam questions sometimes describe exactly this scenario to test whether you notice.

Worked example: binomial probability

A factory tests components with a 0.1 probability of a randomly selected component being defective. In a sample of 8, find P(exactly 2 defective).

Step 1: confirm conditions -- n = 8, p = 0.1, independent trials assumed
Step 2: 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: 8C2 = 28
        P(X = 2) = 28 x 0.01 x 0.531441 ~= 0.149

Show the binomial coefficient (ⁿCₓ) as a distinct step — it allows partial credit if a later arithmetic step slips.

Worked example: cumulative binomial probability using tables

Find P(X <= 3) for X ~ B(10, 0.3) using a cumulative binomial table rather than summing four separate formula calculations.

Step 1: confirm the binomial conditions hold (n = 10, p = 0.3, constant
        and independent)
Step 2: locate the row for n = 10, p = 0.3 in the cumulative binomial
        table
Step 3: read off P(X <= 3) directly as the tabulated value
        P(X <= 3) = 0.6496

The specification explicitly expects fluency with both routes – the full formula calculation for a single value of X, and reading a cumulative table for P(X ≤ x) – since summing several individual terms by hand is slower and more error-prone than reading one table value. Practise recognising which route a question is asking for from its wording: “find the probability that exactly…” usually points to the formula; “find the probability that at most…” or “no more than…” usually points to a cumulative table read.

Deriving the binomial mean and variance conceptually

The specification’s own framing expects more than quoting np and np(1 − p) — it expects you to explain why these follow from the single-trial Bernoulli values. Since a binomial random variable is the sum of n independent Bernoulli trials, and the mean of a sum of independent random variables is the sum of their individual means, E(X) for the binomial is simply n lots of the single-trial mean p, giving np. The same logic extends to variance: because the trials are independent, the variance of the sum equals the sum of the individual variances, giving n lots of p(1 − p), or np(1 − p). Being able to state this derivation in a sentence, not just the final formulas, is what separates a strong answer on this sub-topic from one that has only memorised the result.

Key terms

Bernoulli trial — a single experiment with exactly two outcomes. Bernoulli distribution — the distribution of one Bernoulli trial, mean p, variance p(1 − p). Binomial distribution — the distribution of the number of successes in n independent, identical Bernoulli trials, X ~ B(n, p). Independent trials — outcomes that do not affect one another. ⁿCₓ (n choose x) — the number of ways to arrange x successes among n trials, n! / (x!(n − x)!).

Common mistakes

• Applying the binomial formula without checking all four conditions hold.
• Confusing single-trial Bernoulli mean/variance (p, p(1 − p)) with the full binomial values (np, np(1 − p)).
• Miscalculating or omitting the binomial coefficient ⁿCₓ.
• Forgetting variance is np(1 − p) — standard deviation needs an extra square-root step.

Quick self-test

1. State the four conditions required for a binomial distribution to apply.
2. Give the mean and variance of a single Bernoulli trial with probability p.
3. Give the mean and variance of X ~ B(n, p).
4. Calculate 5C2.
5. Why might sampling without replacement break the binomial model?

Answers: 1. Fixed number of trials n; two outcomes per trial; constant probability p; independent trials. 2. Mean = p, variance = p(1 − p). 3. Mean = np, variance = np(1 − p). 4. 5C2 = 10. 5. It removes independence, since removing an item changes the probability of success for the next trial.

How this connects within the unit

S1.3 builds directly on S1.1 (further probability) and S1.2 (discrete random variables) earlier in Unit PSM1 – the binomial distribution is itself an example of a discrete random variable, and the probability rules from S1.1 are the reason the binomial formula takes the form it does. Reviewing those two sub-topics alongside S1.3 makes the mean and variance derivations above considerably more intuitive than treating the formulas in isolation.

Related resources

• Bernoulli and Binomial Distributions study guide
• Bernoulli and Binomial Distributions practice questions

Official syllabus

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