Year 12 Maths Extension 1 (2027) Statistical analysis

Bernoulli distributions

Q: What is a Bernoulli trial?

A single experiment with two outcomes, success with probability p and failure with probability q equals one minus p.

Q: What are the mean and variance?

The mean is p and the variance is p times one minus p.

Q: When is the variance largest?

At p equals one half, where it equals 0.25.

Q: How many successes are expected over n trials?

n times p, if the trials are independent with the same p.

Q: Why does without replacement matter?

It changes p between draws, so the trials are no longer identically distributed.

20 practice questions 2 video lessons Theory + worked examples

Meet the Bernoulli distribution in NSW Year 12 Mathematics Extension 1. It models a single trial with just two outcomes — success or failure — and is the building block of the binomial distribution.

You will learn to define a Bernoulli random variable, assign the probabilities of success and failure, and use its mean and variance — the starting point for the binomial and sampling work that follows in the Extension 1 course.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

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Watch 2 video(s)

Bernoulli sequence Watch
Introduction to the Bernoulli Distribution Watch

Theory

A Bernoulli trial has two outcomes: success $(X = 1)$ with probability $p$ , failure $(X = 0)$ with probability $q = 1 - p$ . Its mean is $p$ and variance $p (1 - p)$ . This NSW Year 12 Mathematics Extension 1 topic is NESA outcome ME1-12-06.

A Bernoulli trial is a single experiment with exactly two outcomes — success $(X = 1)$ or failure $(X = 0)$ . The random variable $X$ that records the result has a Bernoulli distribution.

For success probability $p$ (and $q = 1 - p$ ): $P (X = 1) = p$ , $P (X = 0) = q$ . Its mean and variance are $E (X) = μ = p$ and $Var (X) = p (1 - p) = p q$ .

Over $n$ repeated independent trials, the expected number of successes is $n p$ .

NESA link. Part of the Year 12 The binomial distribution and sampling distribution of the mean focus area, outcome ME1-12-06 ("solves problems involving binomial distributions, sampling distribution of the mean and the central limit theorem") with MAO-WM-01.

Two outcomes:

P (X = 0) = q

P (X = 1) = p

Var (X) = p (1 - p)

is greatest at

p = 0.5

P (X = 1) = p, P (X = 0) = q = 1 - p .

P(X=1) = p, P(X=0) = q = 1 - p

E (X) = μ = p, Var (X) = p (1 - p) = p q .

E(X) = p, Var(X) = p(1-p) = pq

Single trial. A Bernoulli trial is one trial; several such trials form a binomial experiment. The variance $p (1 - p)$ is largest at $p = 0.5$ .

How to model a Bernoulli trial

Identify the two outcomes (success and failure).
Read off $p$ , the probability of success; then $q = 1 - p$ .
Mean and variance: $E (X) = p$ , $Var (X) = p q$ .
Over $n$ trials, expected successes $= n p$ (if the trials are independent with the same $p$ ).

Example 1 — Rolling a six

A die is rolled and success is a six. Write the distribution and find

E (X)

and

Var (X)

Solution

$p = \frac{1}{6}$ , $q = \frac{5}{6}$ .

$E (X)$	$=$	$p = \frac{1}{6}$
$Var (X)$	$=$	$\frac{1}{6} \cdot \frac{5}{6} = \frac{5}{36}$

E(X) = 1/6, Var(X) = 5/36

$E (X) = \frac{1}{6}$ , $Var (X) = \frac{5}{36}$ .

Example 2 — Given p

A Bernoulli trial has

p = 0.2

. Find the mean and variance.

Solution

$E (X)$	$=$	$0.2$
$Var (X)$	$=$	$0.2 \times 0.8 = 0.16$

mean 0.2, variance 0.16

Mean $0.2$ , variance $0.16$ .

Example 3 — Expected successes

A spinner lands on red with probability

0.25

. If it is spun

60

times, how many reds are expected?

Solution

n p

=

60 \times 0.25 = 15

expected reds = 15

About $15$ reds are expected.

Example 4 — Independence check

Two balls are drawn without replacement from a bag of

3

red and

2

blue. Is each draw a Bernoulli trial with the same

p

Solution

A single draw (red or not) is a Bernoulli trial. But without replacement the probability of red changes after the first draw, so the two draws are not identical, independent trials.

no - draws without replacement are not identically distributed

No — the draws are not identically distributed.

Common pitfalls

One trial only. A Bernoulli trial is a single trial; several form a binomial experiment.

Independence. Repeated trials must be independent with the same

p

— drawing without replacement breaks this.

Variance form.

Var (X) = p q

, largest at

p = 0.5

Mean is just $p$ . For a single Bernoulli trial

E (X) = p

, not

n p

Frequently asked questions

What is a Bernoulli trial?

A single experiment with two outcomes, success $(X = 1)$ with probability $p$ and failure $(X = 0)$ with probability $q = 1 - p$ .

What are the mean and variance?

$E (X) = p$ and $Var (X) = p (1 - p) = p q$ .

When is the variance largest?

At $p = 0.5$ , where $p (1 - p) = 0.25$ .

How many successes are expected over n trials?

$n p$ , if the trials are independent with the same $p$ .

Why does 'without replacement' matter?

It changes $p$ between draws, so the trials are no longer identically distributed.

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Binomial distributions

Bernoulli distributions

📖 Theory

How to model a Bernoulli trial

Common pitfalls

Frequently asked questions

Theory