Characteristics of a binomial distribution; the notation B(n, p); the conditions for a binomial model; the mean np; calculating binomial probabilities for n up to 10.

What this dot point is asking

Edexcel Higher tier code 3p.11 requires you to know and interpret the characteristics of a binomial distribution, to use the notation $B(n, p)$, to know the conditions that make a binomial model suitable, to use the mean $np$, and to calculate binomial probabilities (questions are not set with $n$ larger than $10$). Probabilities may be found by any standard method (calculator functions, spreadsheets or Pascal's triangle).

What the binomial distribution models

Typical situations are the number of heads in several coin flips, the number of correct answers when guessing, or the number of faulty items in a sample. The notation $X \sim B(n, p)$ records the two parameters: $n$ (the number of trials) and $p$ (the probability of success on each trial).

The conditions for a binomial model

A binomial model is only valid when all four conditions hold:

1. There is a fixed number of trials, $n$.
2. Each trial has two outcomes: success or failure.
3. The probability of success $p$ is constant for every trial.
4. The trials are independent of each other.

Edexcel often asks you to state the conditions or to judge whether a binomial model is appropriate in a context. If, for example, items are drawn without replacement (so $p$ changes), the trials are not independent and the binomial model does not strictly apply.

The mean of a binomial distribution

This formula is one Edexcel expects you to know. For $8$ flips of a fair coin, the mean number of heads is $8 \times 0.5 = 4$; for $20$ items with a $0.1$ fault rate, the expected number of faults is $20 \times 0.1 = 2$. The mean is the long-run average number of successes, not a guaranteed value.

Calculating binomial probabilities

To find the probability of exactly $r$ successes you use the binomial term

where $\binom{n}{r}$ is the number of ways to choose which $r$ of the $n$ trials are successes (read from Pascal's triangle or a calculator). Edexcel allows any standard method, and caps $n$ at $10$ so the calculations stay manageable. For "at least one" questions, the complement is quickest: $P(X \ge 1) = 1 - P(X = 0)$.

The three parts of the term each have a meaning worth understanding. The power $p^r$ is the probability that $r$ particular trials are all successes; the power $(1-p)^{n-r}$ is the probability that the remaining $n - r$ trials are all failures; and the coefficient $\binom{n}{r}$ counts how many different ways those $r$ successes can be arranged among the $n$ trials. Multiplying them gives the total probability of getting exactly $r$ successes in any order. Seeing the term this way helps you avoid the common error of dropping the coefficient, which would only count one specific arrangement.

Recognising a binomial situation

Many exam questions do not say "binomial"; you have to recognise the situation from the four conditions. Look for a fixed number of repeated trials with a clear "success", a constant success probability, and independence. Classic contexts are coin flips, multiple choice guessing, items passing or failing a test, and shots hitting or missing a target. If the context instead involves drawing without replacement, a changing probability, or a count with no fixed maximum, the binomial model is not appropriate, and saying so (with a reason) earns marks just as readily as a calculation.

Shape and interpretation

A binomial distribution is symmetric when $p = 0.5$ and skewed when $p$ is close to $0$ or $1$ (skewed towards the more likely outcome). Knowing the mean $np$ tells you where the distribution is centred, so you can sense-check a probability: most of the probability sits near $np$ successes, and outcomes far from $np$ are unlikely. For example, in $B(10, 0.5)$ the most likely number of successes is around $5$, and getting $0$ or $10$ is rare, which matches everyday intuition about flipping a fair coin ten times.