A random variable is a number produced by chance. Whether you can list its possible values or only give a range is what separates the two kinds, and it decides everything that follows.
Today's hook, Every probability question you have met so far has been about outcomes. Naming a number attached to each outcome turns those outcomes into something you can average, graph and model.
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Recall, your gut answer first
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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Recall, your gut answer first
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Roll two dice and add them. List every value the total could take. Now measure the height of the next person to walk past. Could you list every value that might be?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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Countable list, or an interval
Work through the core explanation before applying it.
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Countable list, or an interval
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A random variable takes its value from the outcome of a random experiment. It is discrete if you can list its possible values, usually because it comes from counting. It is continuous if it can take any value in an interval, usually because it comes from measuring.
discrete: countable list, from counting continuous: any value in an interval, from measuring
Counting against measuring
Number of goals, number of defective items, dice totals: discrete. Height, time, mass, temperature: continuous.
Discrete does not mean small
A discrete variable can have infinitely many values, such as the number of coin tosses until the first head. What matters is that they can be listed.
Measurement precision is not the point
Height is continuous even though a ruler reads to the nearest millimetre. The variable is what is being measured, not what the instrument reports.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
A random variable takes its value from the outcome of a random experiment.
A discrete random variable has values that can be listed, typically from counting.
A continuous random variable can take any value in an interval, typically from measuring.
Capital letters name the variable and lower case its particular values, as in $P(X = x)$.
Understand
Concepts
Why counting produces discrete variables and measuring produces continuous ones.
Why a discrete variable can still have infinitely many possible values.
Why the precision of an instrument does not make a continuous variable discrete.
Can do
Skills
Decide whether a described variable is discrete or continuous, with a reason.
Give practical examples of each kind.
List the possible values of a discrete random variable in a simple experiment.
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Key terms
Random variableA quantity whose value is decided by the outcome of a random experiment. Like this: $X$ is the total when two dice are rolled.
Discrete random variableOne whose possible values can be listed, usually from counting. Like this: the total on two dice can only be 2, 3, 4, ..., 12.
Continuous random variableOne that can take any value in an interval, usually from measuring. Like this: the height of a randomly chosen adult.
Random experimentA procedure with a well-defined set of possible outcomes that can be repeated. Like this: rolling two dice and recording the total.
Possible valuesThe complete set of values a random variable can take. Like this: for two dice the possible values run from 2 to 12 inclusive.
Notation $P(X = x)$The probability that the random variable $X$ takes the particular value $x$. Like this: $P(X = 7) = \frac{6}{36}$ for two dice.
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What a random variable is
Work through the core explanation before applying it.
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What a random variable is
core concept
A random experiment is any repeatable procedure with a well-defined set of outcomes: rolling dice, drawing a card, measuring a plant. A **random variable** attaches a number to each outcome.
Rolling two dice has 36 outcomes. Defining $X$ as the total turns each of them into a number between 2 and 12. The experiment produces the outcome; the variable produces the number.
The convention is a capital letter for the variable and a lower case letter for a particular value, so $P(X = 7)$ reads "the probability that the total is 7".
The variable is a choice, not a property of the experiment. The same two dice could define $X$ as the total, the difference, or the larger of the two. Different variables, same experiment.
Quick check: two dice are rolled and $X$ is the total. What is the smallest value $X$ can take?
A random variable attaches a number to each outcome of a random experiment. Capital letters name the variable, lower case its particular values, and $P(X = x)$ is the probability of a particular value. The choice of variable is separate from the experiment.
Pause, copy the definition, the two-dice example with $X$ as the total, and the $P(X = x)$ notation, into your book.
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Discrete or continuous
core concept
We just saw what a random variable is. That raises a question: are all random variables the same kind of thing? This card answers it → no, and the split between listable and unlistable values is the one that matters.
A variable is **discrete** if its possible values can be listed. The total on two dice can only be $2, 3, 4, \ldots, 12$, so it is discrete. Counting almost always produces a discrete variable.
A variable is **continuous** if it can take any value in an interval. A person's height could be 170 cm, or 170.4 cm, or 170.41 cm, with no gaps between the possibilities. Measuring almost always produces a continuous variable.
The quick test is: **did you count it or measure it?** Number of goals, faulty items, students present are counted and therefore discrete. Height, mass, time, temperature are measured and therefore continuous.
Discrete does not mean finite. The number of coin tosses until the first head could be 1, 2, 3 and so on with no upper limit, yet those values can still be listed in order. Listability is the test, not how many there are.
Which of these is NOT a discrete random variable?
Discrete means the possible values can be listed, which counting produces. Continuous means any value in an interval, which measuring produces. Ask whether the quantity was counted or measured. Discrete does not mean finite, only listable.
Pause, copy both definitions, the count-or-measure test with three examples of each, and the note that discrete can be infinite, into your book.
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Why the distinction decides what comes next
core concept
We just saw how to classify a variable. That raises a question: why does the course care which kind it is? This card answers it → because the two kinds are handled by completely different machinery later.
For a **discrete** variable you can give each value its own probability, and those probabilities add to 1. Asking for $P(X = 7)$ is a sensible question with a non-zero answer.
For a **continuous** variable, asking for the probability of an exact value gives zero: there are infinitely many possible heights, so no single one has positive probability. You ask for the probability of a **range** instead, such as $P(170 < X < 180)$.
That is why Year 12 handles the two separately, with probability distributions for discrete variables and probability density functions for continuous ones. Classifying correctly at this stage decides which toolkit applies.
Measurement precision is a red herring. Height reported to the nearest centimetre still describes a continuous variable. What matters is the quantity being measured, not the resolution of the instrument recording it.
Fill the blank: a variable produced by measuring rather than counting is usually .
Discrete variables give each value its own probability, summing to 1. Continuous variables give zero probability to any exact value, so you ask about ranges instead. Classifying correctly decides which methods apply later, and instrument precision does not change the classification.
Pause, copy why $P(X = x)$ is useful for discrete variables and zero for continuous ones, and the note that measurement precision does not change the type, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · LISTING THE VALUES
Two dice are rolled and $X$ is the total. State the type of random variable and list its possible values.
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$X$ counts spots, so it is discrete
Counted, not measured.
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The smallest total is $1 + 1 = 2$; the largest is $6 + 6 = 12$
Consider the extremes.
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$X$ can take $2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12$: eleven possible values
Every whole number in between is achievable.
PROBLEM 2 · CLASSIFYING
Classify each as discrete or continuous, with a reason: (a) the number of students absent, (b) the time to run 100 m, (c) the mass of an apple.
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(a) Discrete: absences are counted, and the values are $0, 1, 2, \ldots$
Counting gives a listable set.
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(b) Continuous: time is measured, so $t$ can take any value in an interval
No gaps between possible times.
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(c) Continuous: mass is measured, so any value in a range of $\text{kg}$ is possible
A scale reading to one gram does not change this.
PROBLEM 3 · AN INFINITE DISCRETE VARIABLE
A coin is tossed until the first head. $X$ is the number of tosses required. Is $X$ discrete or continuous?
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The possible values are $1, 2, 3, \ldots$ with no upper limit
There is no largest number of tosses.
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But $1, 2, 3, \ldots$ can still be listed in order
Listability, not finiteness, is the test.
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$X$ is discrete, with infinitely many possible values
Discrete does not mean a finite list.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
+10 XP
Is the number of cars passing in an hour discrete or continuous?
Is the temperature at noon discrete or continuous?
List the possible values when one die is rolled and $X$ is the number shown.
Give one practical example of each type.
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Revisit your two experiments
Run the quick drill and copy the summary into your book.
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Revisit your two experiments
At the start you listed the possible totals for two dice, then tried to list possible heights. Name the type of each variable, and explain in one sentence what made the second list impossible.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
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Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
UnderstandBand 33 marks
Q1. Define a random variable, and explain the difference between a discrete and a continuous random variable. (3 marks)
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ApplyBand 33 marks
Q2. Classify each of the following, giving a reason: (a) the number of text messages you send today, (b) the volume of water in a bottle, (c) the number of heads in ten coin tosses. (3 marks)
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UnderstandBand 42 marks
Q3. A student says the height of a person is discrete because a ruler only measures to the nearest millimetre. Explain why this reasoning is wrong. (2 marks)
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📖 Comprehensive answers (click to reveal)
Practice 1: discrete, cars are counted. Practice 2: continuous, temperature is measured. Practice 3: 1, 2, 3, 4, 5, 6. Practice 4: for example the number of goals in a match (discrete) and the mass of a parcel (continuous).
Q1 (3 marks): A random variable is a variable whose value is determined by the outcome of a random experiment [1]. A discrete random variable has possible values that can be listed, and typically arises from counting [1]. A continuous random variable can take any value within an interval, and typically arises from measuring [1].
Q2 (3 marks): (a) Discrete, because messages are counted and the values are whole numbers that can be listed [1]. (b) Continuous, because volume is measured and can take any value in a range [1]. (c) Discrete, because the number of heads is counted and can only be $0, 1, 2, \ldots, 10$ [1].
Q3 (2 marks): The classification depends on the quantity itself, not on the instrument used to record it: height can in principle take any value in an interval, with no gaps between possible values [1]. Rounding a measurement to the nearest millimetre is a limitation of the ruler, and the underlying variable remains continuous no matter how precisely it is reported [1].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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Boss battle · Variable Vetting
earn bronze · silver · gold
Classify random variables as discrete or continuous and list their possible values. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.