Discrete probability distributions: WACE Year 12 Mathematics Methods Unit 3

Syllabus dot point

“Define discrete random variables and construct and verify discrete probability distributions, including finding unknown probabilities”

8 min readUpdated 2 Jun 2026

Quick answer

A discrete random variable takes separate, countable values. A valid probability distribution lists each value with a probability between $0$ and $1$ , and all the probabilities sum to exactly $1$ . Setting the sum to $1$ is how you find an unknown probability.

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What this dot point is asking
Discrete random variables
Finding an unknown probability
Probabilities of events
The discrete uniform distribution
Distributions given by a rule
Cumulative distribution
Exam-style questions

What this dot point is asking

SCSA Unit 3 introduces discrete random variables as the foundation for the binomial distribution and all of probability in the course. This dot point asks you to define such variables and build and verify their probability distributions, examined in both sections of the WACE written examination.

Discrete random variables

A random variable assigns a number to each outcome of a random process. A discrete random variable takes values that can be listed separately, such as $0,1,2,3$ , typically counts. This contrasts with a continuous variable, which takes any value in an interval.

Discrete probability distribution

A discrete probability distribution lists every value $x$ the variable can take alongside its probability $P(X=x)$ . It is valid when both conditions hold:

0\le P(X=x)\le 1 \text{ for every } x, \qquad \sum_x P(X=x)=1.

The second condition reflects certainty: one of the listed outcomes must occur, so the probabilities exhaust all possibilities.

Finding an unknown probability

The most common SCSA task is a distribution with one probability written as an unknown. Because the probabilities must sum to $1$ , you set up an equation and solve.

Finding an unknown probability

A random variable $X$ has the distribution below with one probability written as an unknown $k$ .

$x$	$1$	$2$	$3$	$4$
$P(X=x)$	$0.1$	$0.3$	$k$	$0.2$

Step 1: Set up the sum-to-one equation

Because one of the four outcomes must occur, all probabilities must total exactly $1$ . Writing out the sum gives the equation to solve.

0.1 + 0.3 + k + 0.2 = 1.

Step 2: Solve for $k$

Collect the known values on the left side and isolate $k$ .

0.6 + k = 1 \;\Rightarrow\; k = 0.4.

Step 3: Check validity

Confirm that $k$ satisfies both distribution conditions: $0\le 0.4\le 1$ , so the value is a legitimate probability, and the probabilities still sum to $1$ with this value in place.

Final answer: $k=0.4$ . Markers reward the sum-to-one equation and the validity check.

Probabilities of events

Once the distribution is known, the probability of a compound event is the sum of the probabilities of the values it includes. For the table above, $P(X\ge 3)=P(X=3)+P(X=4)=0.4+0.2=0.6$ , and $P(X<3)=1-0.6=0.4$ by the complement.

The discrete uniform distribution

When all $n$ outcomes are equally likely, each has probability $\dfrac{1}{n}$ ; this is the discrete uniform distribution. A fair die gives $P(X=x)=\dfrac{1}{6}$ for $x=1,\dots,6$ . The uniform case is the simplest distribution and a useful check that probabilities sum to $1$ .

A probability from a uniform distribution

A fair eight-sided die

Find the probability of rolling a number greater than $5$ .

Each face has probability $\dfrac{1}{8}$ . The favourable outcomes are $6,7,8$ , three values.

P(X>5) = 3\times\frac{1}{8} = \frac{3}{8}.

So the probability is $0.375$ .

Distributions given by a rule

Sometimes the probabilities are not listed in a table but defined by a formula such as $P(X=x)=cx$ or $P(X=x)=c(x+1)$ for stated values of $x$ . The method is the same: write the sum of all the probabilities, set it equal to $1$ , and solve for the constant.

Finding the constant in a rule-based distribution

A discrete random variable has $P(X=x)=cx^{2}$ for $x=1,2,3$ and zero otherwise. We find $c$ so that the distribution is valid.

Step 1: Write the sum-to-one equation using the rule

Substitute each value of $x$ into the formula and sum all probabilities, then set the total equal to $1$ .

c(1)^{2}+c(2)^{2}+c(3)^{2}=c(1+4+9)=14c=1.

The factor of $c$ is common to every term, so the sum simplifies by factoring before equating.

Step 2: Solve for the constant

Divide both sides by $14$ .

c=\frac{1}{14}.

Step 3: State the resulting distribution

Substitute $c=\dfrac{1}{14}$ back into the rule to obtain the complete distribution.

P(X=1)=\frac{1}{14}, \quad P(X=2)=\frac{4}{14}, \quad P(X=3)=\frac{9}{14}.

Final answer: $c=\dfrac{1}{14}$ , giving $P(X=1)=\dfrac{1}{14}$ , $P(X=2)=\dfrac{4}{14}$ , $P(X=3)=\dfrac{9}{14}$ . These sum to $1$ as required. Markers reward forming the sum from the rule, equating to $1$ , and solving for the constant.

Cumulative distribution

The cumulative probability $P(X\le x)$ adds all probabilities up to and including $x$ , increasing from $0$ to $1$ as $x$ runs through the values. It is the discrete analogue of the area under a density curve and is the quantity the calculator returns for at-most events. Reading off $P(X\le x)$ lets you answer at-least and between questions by subtraction.

Common errors

Probabilities not summing to one: Always verify $\sum P(X=x)=1$ ; an unknown is usually found from this equation.
Allowing a probability outside $[0,1]$: A negative probability or one above $1$ means an arithmetic error; check before continuing.
Confusing $P(X\ge 3)$ with $P(X>3)$: The first includes $X=3$ ; the second does not.
Treating a continuous quantity as discrete: Counts are discrete; measurements such as height or time are continuous and need a density function instead.

Exam-style questions

Questions in the style of SCSA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

WACE 2022-style6 marks

Calculator-assumed. A discrete random variable

X

has

P(X=x)=cx

for

x=1,2,3,4

and zero otherwise. (a) Find the value of

c

. (b) Find

P(X\ge 3)

. (c) Find

E(X)

Show worked answer →

A build-and-verify question with an unknown constant.

(a) The probabilities must sum to $1$ : $c(1)+c(2)+c(3)+c(4)=10c=1$ , so $c=\dfrac{1}{10}=0.1$ .

(b) $P(X\ge 3)=P(X=3)+P(X=4)=0.3+0.4=0.7$ .

Markers reward the sum-to-one equation for $c$ , the correct cumulative sum, and the weighted mean.

WACE 2024-style4 marks

Calculator-free. The table gives a probability distribution:

P(X=0)=0.2

P(X=1)=k

P(X=2)=0.3

P(X=3)=0.1

. (a) Find

k

. (b) Find

P(X\ge 2)

Show worked answer →

A calculator-free verification and event probability.

(a) Sum to one: $0.2+k+0.3+0.1=1$ , so $k=0.4$ . Check $0\le 0.4\le 1$ , valid.

(b) $P(X\ge 2)=P(X=2)+P(X=3)=0.3+0.1=0.4$ .

Markers reward the sum-to-one equation, the validity check, and the correct event sum including $X=2$ .

Discrete probability distributions: WACE Year 12 Mathematics Methods Unit 3

What this dot point is asking

Discrete random variables

Finding an unknown probability

Step 1: Set up the sum-to-one equation

Step 2: Solve for $k$

Step 3: Check validity

Probabilities of events

The discrete uniform distribution

A fair eight-sided die

Distributions given by a rule

Step 1: Write the sum-to-one equation using the rule

Step 2: Solve for the constant

Step 3: State the resulting distribution

Cumulative distribution

Exam-style questions

Practise this

Sources & how we know this

What this dot point is asking

Discrete random variables

Finding an unknown probability

Step 1: Set up the sum-to-one equation

Step 2: Solve for kkk

Step 3: Check validity

Probabilities of events

The discrete uniform distribution

A fair eight-sided die

Distributions given by a rule

Step 1: Write the sum-to-one equation using the rule

Step 2: Solve for the constant

Step 3: State the resulting distribution

Cumulative distribution

Exam-style questions

Practise this

Sources & how we know this

Step 2: Solve for $k$