Year 10: Understanding probability

Probability is all about how likely something is to happen. We use it every day, often without realising it:

“What’s the chance it will rain today?”
“How likely are we to win this game?”
“What’s the chance of getting heads if I flip this coin?”

In mathematics, probability gives us a precise way to talk about that “chance”. In this article you’ll learn key ideas and language used in Year 10 probability: sample space, trials, simple events, compound events, and how to use Venn diagrams.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

Basic language of probability

Experiment and trial

A random experiment is any situation where:

you know what could happen in theory, but
you don’t know for sure what will happen this time.

A single performance of that experiment is called a trial.

Example:

Flipping a coin once = 1 trial of the “coin flip” experiment.
Rolling a die 10 times = 10 trials of the “roll a die” experiment.
Picking a student at random from your class = 1 trial of the “pick a student” experiment.

Sample space and outcomes

The sample space is the set of all possible outcomes of an experiment.

Example: Rolling a fair six-sided die

Outcomes: 1, 2, 3, 4, 5, 6
Sample space: $S = {1, 2, 3, 4, 5, 6}$

An event is a collection of outcomes that we’re interested in. Events are usually written as sets.

Example: Event “rolling an even number”

Outcomes that are even: 2, 4, 6
Event: $E = {2, 4, 6}$

The probability of an event

For simple situations where all outcomes are equally likely, we use:

Probability = Number of favourable outcomes ÷ Total number of outcomes

Example: Probability of rolling a 3 on a fair die

Sample space $S = {1, 2, 3, 4, 5, 6}$ (6 outcomes).
Event “roll a 3” has 1 favourable outcome: {3}.

So, $\Pr (3) = 1 / 6$ .

Probabilities are numbers between 0 and 1 (or 0% and 100%):

0 means “impossible”.
1 means “certain”.
Values in between show different levels of likelihood.

Skill quiz: Identify sample spaces and events

Practise distinguishing the full set of outcomes from the event of interest.

Sample spaces and event language

Simple events

A simple event is an event that involves just a single step or a single outcome type. There is only one “thing” happening.

Examples of simple events:

Flipping a coin once and getting heads.
Rolling a die once and getting a 5.
Selecting one card from a deck and getting a heart.

Working with simple events

For simple events with equally likely outcomes, we still use:

Pr(event) = Number of favourable outcomes ÷ Total number of outcomes

Example: Drawing a red ball

A bag contains 3 red balls and 5 blue balls. One ball is drawn at random.

Total outcomes = 3 + 5 = 8.
Favourable outcomes for “red” = 3.

$\Pr (red) = 3 / 8$ .

Skill quiz: Count favourable outcomes

Count the favourable outcomes and total equally likely outcomes before forming the probability fraction.

Probability of simple events

Compound events

A compound event involves more than one step or more than one condition. For example:

Flipping a coin twice.
Rolling two dice.
Drawing two cards from a deck (with or without replacement).

Sample space for two steps

When there are two steps, you can think in terms of ordered pairs, like: (outcome of first step, outcome of second step).

Example: Flipping a coin twice

Each flip can be Heads (H) or Tails (T).

Sample space:

HH (heads then heads)
HT (heads then tails)
TH (tails then heads)
TT (tails then tails)

So, $S = {HH, HT, TH, TT}$ .

“AND” and “OR” in compound events

When dealing with compound events, you’ll often see the words AND and OR:

A AND B means both events A and B happen at the same time. (This is an intersection of events.)
A OR B means A happens, or B happens, or both. (This is a union of events.)

Example: Rolling two dice

Let A = event “the first die shows a 4”.
Let B = event “the total of the two dice is 7”.
A AND B means “first die is 4 AND total is 7”.
A OR B means “first die is 4 OR total is 7 (or both)”.

Independent events

Two events are independent if one does not affect the probability of the other.

Example: Flip a coin and roll a die

Event A: “coin shows heads”.
Event B: “die shows a 6”.

What happens on the coin doesn’t change what happens on the die, so A and B are independent.

For independent events A and B, the probability of both happening (A AND B) is:

Pr(A and B) = Pr(A) × Pr(B)

Example: Heads and a 6

Pr(heads) = 1/2
Pr(6) = 1/6

So, $\Pr (heads and 6) = 1 / 2 \times 1 / 6 = 1 / 12$ .

Skill quizzes: Combine events and experiments

List combined outcomes, then practise multiplying probabilities when the events are independent.

Venn diagrams in probability

Venn diagrams are used to visually show how events overlap. They are especially useful for “A AND B”, “A OR B” and “not A” (the complement of A).

Basic structure

A Venn diagram usually looks like this:

A rectangle representing the universal set (all outcomes in the sample space).
Circles inside for each event (e.g. one circle for event A, another for event B).

Example: Students in a Year 10 class

Let’s say:

Event A: students who play soccer.
Event B: students who play basketball.

Some students might be in both circles (they play both sports). Some may be outside both circles (they play neither).

Union (A OR B)

The union of A and B (written as $A \cup B$ ) includes all outcomes that are in A, or in B, or in both.

On a Venn diagram, this is the area covered by either circle.

Probability formula for union (general form):

$\Pr (A \cup B) = \Pr (A) + \Pr (B) - \Pr (A \cap B)$

We subtract $\Pr (A \cap B)$ (the intersection) because it’s counted twice in $\Pr (A) + \Pr (B)$ .

Intersection (A AND B)

The intersection of A and B (written as $A \cap B$ ) includes only outcomes that are in both A and B.

On the Venn diagram, this is where the two circles overlap.

Example: Students who play both sports

If 12 students play soccer, 10 students play basketball and 5 students play both, then:

A = “plays soccer”, B = “plays basketball”.
|A| = 12, |B| = 10, |A ∩ B| = 5.

The number who play soccer or basketball (or both) is:
$| A \cup B | = | A | + | B | - | A \cap B | = 12 + 10 - 5 = 17$ .

Complement (Not A)

The complement of A (written as $A'$ or $A^{c}$ ) means “not A” – all outcomes that are outside event A.

On the Venn diagram, this is everything in the rectangle that is not in circle A.

We have the useful relationship:

Pr(not A) = 1 − Pr(A)

Example: Complement

If the probability that a student catches the bus to school is 0.4, then:

$\Pr (does not catch bus) = 1 - 0.4 = 0.6$ .

Skill quizzes: Read unions, intersections and complements

Use the two circles to identify overlaps and outside regions, then connect the diagram with a two-way table.

Bringing it all together

Let’s summarise the key terms:

Trial – one performance of a random experiment.
Sample space – the set of all possible outcomes.
Event – a set of outcomes we care about.
Simple event – involves one step or outcome type.
Compound event – involves multiple steps and/or conditions.
Venn diagram – a visual tool to show how events overlap.
Union (A ∪ B) – A or B or both.
Intersection (A ∩ B) – A and B at the same time.
Complement (A') – not A.

Understanding these ideas sets you up for more advanced probability topics in Year 10 and beyond, including tree diagrams, conditional probability and real-life applications in science, finance, and everyday decision-making.

Skill quizzes: Connect event language and calculations

Review sample spaces and simple probabilities before tackling compound events.