The binomial distribution: definition, probability mass function, mean and variance

What this dot point is asking

NESA wants you to recognise scenarios that fit the binomial model (fixed number of independent Bernoulli trials with the same success probability), state the probability mass function, and compute the mean, variance and standard deviation.

The answer

Definition

Let $X$ be the number of successes in $n$ independent Bernoulli trials, each with success probability $p$. Then $X$ has the binomial distribution $B(n, p)$ (or $\text{Bin}(n, p)$).

The parameters are:

• $n$: the number of trials (a positive integer).
• $p$: the success probability on each trial ($0 \le p \le 1$).
• $q = 1 - p$: the failure probability.

The probability mass function

For $k = 0, 1, 2, \dots, n$,

The binomial coefficient $\binom{n}{k}$ counts the number of ways to arrange $k$ successes among $n$ trials.

Mean and variance

By summing $n$ independent Bernoulli trials (each with mean $p$ and variance $p(1 - p)$):

Conditions for the binomial model

Four conditions must hold:

1. A fixed number of trials, $n$.
2. Each trial has exactly two outcomes (success or failure).
3. The trials are independent.
4. The probability of success is the same for every trial.

If any of these fails, the distribution is not binomial.

Common scenarios

• Number of heads in $n$ coin flips: $X \sim B(n, p)$ with $p$ the heads probability.
• Number of items defective in a batch of $n$ (with replacement, or with a large enough population to treat as approximately independent).
• Number of correct answers on a multiple-choice test if every question is guessed.
• Number of successful free throws in a fixed number of attempts (independence is a simplifying assumption).

Symmetric and skewed cases

• If $p = \tfrac{1}{2}$, the distribution is symmetric around $\frac{n}{2}$.
• If $p < \tfrac{1}{2}$, the distribution is right-skewed (long tail to the right).
• If $p > \tfrac{1}{2}$, it is left-skewed.

The skew decreases as $n$ grows; for large $n$, the binomial approaches the normal (see the normal approximation dot point).

Building one distribution, stage by stage

To see where the pmf, the mean and the variance come from, take a concrete case: a biased coin with $p = 0.4$ flipped $n = 10$ times, so $X \sim B(10, 0.4)$. The three panels build the whole distribution from the formula and then read its centre and spread off the picture.

Stage 1, compute a few probabilities from the formula. Each bar height is $P(X = k) = \binom{10}{k}(0.4)^k(0.6)^{10 - k}$. For the first few values, $P(X = 0) = (0.6)^{10} \approx 0.006$, $P(X = 1) = \binom{10}{1}(0.4)(0.6)^9 \approx 0.040$, and $P(X = 2) = \binom{10}{2}(0.4)^2(0.6)^8 \approx 0.121$. Each is one number from the same formula.

Stage 2, fill in every $k$ to get the whole distribution. Repeating the formula for $k = 0, 1, \dots, 10$ gives all eleven bars. Because the eleven probabilities are the terms of $(0.6 + 0.4)^{10}$, they sum to $1^{10} = 1$: the distribution is complete and valid.

Stage 3, read the centre and spread. The distribution is centred on its mean $np = 10 \times 0.4 = 4$, marked by the dashed line, and its spread is governed by the variance $np(1 - p) = 10 \times 0.4 \times 0.6 = 2.4$ (standard deviation $\sqrt{2.4} \approx 1.55$). With $p < \tfrac{1}{2}$ the tail leans to the right, exactly as the skew rule predicts.

How exam questions ask about the binomial distribution

The binomial is one of the most reliable sources of marks in Extension 1 statistics, and the question wordings are stable:

• "Let $X$ be the number of [successes] in $n$ [trials] ... find $E(X)$, $\text{Var}(X)$ and the standard deviation": identify $n$ and $p$, then quote $np$, $np(1 - p)$ and its square root. State the model $X \sim B(n, p)$ first.
• "Find the probability of exactly $k$ [successes]": substitute into $\binom{n}{k} p^k (1 - p)^{n - k}$ and evaluate. Showing the substituted formula earns the method mark even before you reach the calculator.
• "Explain why $X$ is (or is not) binomial": name the four conditions, fixed $n$, two outcomes, independence, constant $p$, and say which one fails. Sampling without replacement is the classic failure (the model becomes hypergeometric).
• "Let $Y = aX + b$ ... find $E(Y)$ and $\text{Var}(Y)$": apply $E(aX + b) = aE(X) + b$ and $\text{Var}(aX + b) = a^2 \text{Var}(X)$; the constant $b$ never touches the variance.
• Distribution-shape parts: a sketch or a "describe the shape" cue wants you to mark the mean $np$ and call the shape symmetric (if $p = \tfrac{1}{2}$), right-skewed ($p < \tfrac{1}{2}$) or left-skewed ($p > \tfrac{1}{2}$).