The binomial distribution (QCE Mathematical Methods Unit 3)

What this dot point is asking

QCAA wants you to recognise when a count is binomially distributed, calculate binomial probabilities (by hand for small $n$, by graphics calculator for larger $n$), and apply the mean and variance formulas. The binomial distribution is the most heavily examined probability model in QCE Mathematical Methods Unit 3, and it appears in IA1 PSMTs, IA2 short and extended response, and most EA Paper 2 probability questions.

The answer

When is $X$ binomial: the BINS conditions

A random variable $X$ has a binomial distribution if all four conditions hold.

• B (Binary): Each trial has two outcomes labelled success or failure.
• I (Independent): Trials are independent of one another.
• N (Number fixed): The number of trials $n$ is fixed in advance.
• S (Same probability): The probability of success $p$ is the same on every trial.

If all four hold, write $X \sim \mathrm{Bin}(n, p)$, where $X$ is the number of successes in the $n$ trials.

The binomial probability formula

For $X \sim \mathrm{Bin}(n, p)$ and $k = 0, 1, 2, \ldots, n$,

The binomial coefficient $\binom{n}{k} = \dfrac{n!}{k! \, (n - k)!}$ counts the number of ways to arrange $k$ successes among $n$ trials.

Paper 1 expects this formula by hand for small $n$ (typically $n \leq 6$) with values that simplify to clean fractions.

Paper 2 expects you to use your graphics calculator for any $n$ above about 6, calling functions usually named $\mathrm{binompdf}$ for $P(X = k)$ and $\mathrm{binomcdf}$ for $P(X \leq k)$ (some models take a lower and upper bound instead). Check the exact syntax for your calculator. QCAA allows a graphics calculator with no CAS functionality (and/or a scientific calculator) in Paper 2, so confirm your model against QCAA's graphics calculator list; Paper 1 allows no calculator at all.

Mean and variance

For $X \sim \mathrm{Bin}(n, p)$,

Standard deviation: $\sigma = \sqrt{n p (1 - p)}$.

These come from the fact that a binomial is the sum of $n$ independent Bernoulli trials, each with mean $p$ and variance $p (1 - p)$, and the rules $E(X + Y) = E(X) + E(Y)$ and $\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$ for independent variables.

Cumulative probabilities

QCAA commonly asks for $P(X \leq k)$, $P(X \geq k)$ or $P(a \leq X \leq b)$ rather than a single $P(X = k)$.

• $P(X \leq k) = \sum_{i = 0}^{k} P(X = i)$.
• $P(X \geq k) = 1 - P(X \leq k - 1)$ (use the complement to save work).
• $P(a \leq X \leq b) = P(X \leq b) - P(X \leq a - 1)$.

For $n$ above about 6, do all four with your graphics calculator. For Paper 1 with small $n$, set up the sums explicitly and evaluate by hand.