NSWMaths Extension 1Syllabus dot point

What is a Bernoulli trial, and what are its mean and variance?

Define a Bernoulli random variable, compute its mean and variance, and recognise scenarios that fit the model

A focused answer to the HSC Maths Extension 1 dot point on Bernoulli trials. The definition, mean $p$ , variance $p (1 - p)$ , and the role of Bernoulli trials as the building block of the binomial distribution.

Generated by Claude OpusReviewed by Better Tuition Academy6 min answerUpdated 2026-05-20

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What this dot point is asking

NESA wants you to recognise a Bernoulli trial (a single experiment with two outcomes), state its probability mass function, compute its mean and variance, and see it as the unit of a sequence used to build the binomial distribution.

The answer

Definition

A Bernoulli trial is a random experiment with exactly two possible outcomes, conventionally labelled "success" (value $1$ ) and "failure" (value $0$ ), where success occurs with probability $p$ and failure with probability $q = 1 - p$ .

If $X$ is a Bernoulli random variable with parameter $p$ (written $X \sim B(p)$ or $\text{Bern}(p)$ ):

P(X = 1) = p, \qquad P(X = 0) = 1 - p.

Mean (expected value)

E(X) = 1 \cdot p + 0 \cdot (1 - p) = p.

The expected value of a single Bernoulli trial is its success probability.

Variance

E(X^2) = 1^2 \cdot p + 0^2 \cdot (1 - p) = p.

\text{Var}(X) = E(X^2) - [E(X)]^2 = p - p^2 = p(1 - p).

Equivalently, $\sigma = \sqrt{p (1 - p)}$ .

Standard examples

Flipping a fair coin with $X = 1$ if heads: $p = \tfrac{1}{2}$ .
Rolling a die and checking for a $6$ : $p = \tfrac{1}{6}$ .
Asking a random voter if they support a policy with $p$ unknown.
A medical test giving a positive result on someone with the condition (sensitivity).

Why this matters

A Bernoulli trial is the simplest non-trivial random variable. The binomial distribution counts the number of successes in a fixed number of independent Bernoulli trials.

A sequence of $n$ independent identically distributed Bernoulli trials, where $X_i = 1$ if the $i$ -th trial is a success and $0$ otherwise, has total $S = X_1 + X_2 + \dots + X_n \sim B(n, p)$ .

By linearity of expectation, $E(S) = n p$ . By independence and the variance addition rule, $\text{Var}(S) = n p (1 - p)$ .

Worked example

Direct computation

A biased coin lands heads with probability $0.3$ . Let $X$ be $1$ if heads, $0$ otherwise. Find $E(X)$ and $\text{Var}(X)$ .

$E(X) = 0.3$ . $\text{Var}(X) = 0.3 \cdot 0.7 = 0.21$ . $\sigma = \sqrt{0.21} \approx 0.458$ .

Maximising variance

For what value of $p$ is $\text{Var}(X) = p(1 - p)$ maximised?

This is a quadratic in $p$ with vertex at $p = \tfrac{1}{2}$ . Maximum variance is $\tfrac{1}{4}$ .

A fair Bernoulli trial (coin flip) has the highest variance.

Sum of independent Bernoullis

Three independent fair coin flips. Let $S$ be the total number of heads. Find $E(S)$ and $\text{Var}(S)$ .

Each flip is Bernoulli with $p = \tfrac{1}{2}$ . $S \sim B(3, \tfrac{1}{2})$ .

$E(S) = 3 \cdot \tfrac{1}{2} = \tfrac{3}{2}$ . $\text{Var}(S) = 3 \cdot \tfrac{1}{2} \cdot \tfrac{1}{2} = \tfrac{3}{4}$ .

Identify the model

Is "the time until the next bus arrives" a Bernoulli trial? No, because it has a continuous range of outcomes, not two.

Is "did the bus arrive on time?" a Bernoulli trial? Yes, two outcomes (on time / not on time).

Linearity check

Two independent Bernoulli trials with $p_1 = 0.4$ and $p_2 = 0.6$ . Let $S = X_1 + X_2$ . Find $E(S)$ and $\text{Var}(S)$ .

$E(S) = 0.4 + 0.6 = 1.0$ .

$\text{Var}(S) = 0.4 \cdot 0.6 + 0.6 \cdot 0.4 = 0.24 + 0.24 = 0.48$ .

Note: $S$ is not a binomial because the two trials have different $p$ . But the mean and variance still add by linearity and independence.