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Probability is the maths of chance — how likely something is to happen. This guide walks you through the probability scale from 0 to 1, the language of chance, how to calculate the probability of simple events, and how to list every possible outcome.
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
Every probability sits on a scale: 0 means impossible, 1 means certain.
Probability is a way of measuring how likely something is to happen. Every probability is a number on a scale from 0 to 1.
Words like likely, unlikely and even chance describe probability in everyday language.
Before you put numbers to probability, it helps to describe chance in words. These words match positions on the probability scale:
An outcome is one possible result; an event can be made of several outcomes.
To calculate probability, you need two clear ideas.
An outcome is a single possible result of an experiment. When you roll an ordinary dice, the possible outcomes are 1, 2, 3, 4, 5 and 6 — six outcomes in total.
An event is the thing you are interested in. It can be a single outcome, or a group of outcomes. "Rolling a 4" is an event made of one outcome. "Rolling an even number" is an event made of three outcomes — 2, 4 and 6.
Outcomes are equally likely when each has the same chance — every face of a fair dice, for example. This is the situation you need before you can use the probability calculation in the next section.
For equally likely outcomes, divide favourable outcomes by total outcomes.
When all outcomes are equally likely, there is a simple formula for the probability of an event:
A favourable outcome is one that counts as the event happening.
List outcomes systematically so none are missed or repeated.
To find the total number of outcomes, you often need to list them all. The secret is to be systematic — work in a fixed order so nothing is missed and nothing is counted twice.
Suppose you flip a coin and roll a dice. List the outcomes by taking each coin result in turn and pairing it with every dice number:
There are 12 outcomes in total. A table like this — or an ordered list — guarantees you have found them all.
Once the full list is written, calculating a probability is easy. The probability of getting heads and an even number is , because three outcomes (H,2 / H,4 / H,6) are favourable out of 12.
Probability is the foundation of predicting and reasoning about chance.
Understanding probability now sets you up for plenty of maths ahead — and for thinking clearly about the world:
Whenever you hear "there is a 70% chance of rain" or "you are unlikely to win", you now know exactly what those statements mean — and how someone worked them out.
Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 7). Last reviewed 2026-05-19.
Step-by-step solutions for probability, written exactly the way a tutor would explain them at the board.
Question
Describe each event as impossible, unlikely, even chance, likely or certain: (a) a fair coin lands on tails, (b) rolling a number bigger than 0 on an ordinary dice, (c) rolling a 9 on an ordinary dice.
Step-by-step solution
Step 1
(a) A coin has two equally likely sides, so tails has an even chance.
Step 2
(b) Every face of a dice (1 to 6) is bigger than 0, so this is certain.
Step 3
(c) A dice only has faces 1 to 6, so rolling a 9 is impossible.
Answer
(a) Even chance (b) Certain (c) Impossible.
Question
A fair dice is rolled once. What is the probability of rolling a 3? Give your answer as a fraction.
Step-by-step solution
Step 1
There is only one favourable outcome — rolling a 3.
Step 2
There are 6 outcomes in total: 1, 2, 3, 4, 5 and 6.
Step 3
Probability = favourable outcomes ÷ total outcomes.
Question
A bag holds 4 green counters, 6 yellow counters and 10 purple counters. One counter is taken out without looking. What is the probability it is yellow? Give your answer as a fraction in its simplest form.
Step-by-step solution
Step 1
Find the total number of counters by adding them all up.
Question
A spinner has the colours red and blue. A coin has heads and tails. The spinner is spun and the coin is flipped. List all the possible outcomes, then state how many there are.
Step-by-step solution
Step 1
Be systematic: take the first spinner colour, red, and pair it with each coin result.
Question
Two fair coins are flipped together. (a) List all the possible outcomes. (b) Work out the probability of getting exactly one head.
Step-by-step solution
Step 1
(a) Be systematic. Take the first coin as Heads, then pair with each result of the second coin, then repeat with Tails.
The important words for probability and what they mean — learn these so you can explain your thinking clearly.
A measure of how likely an event is to happen, given as a number from 0 to 1.
A line from 0 to 1 used to show how likely events are. 0 is impossible and 1 is certain.
A single possible result of an experiment.
Example
Rolling a 5 is one outcome of rolling a dice.
The result you are interested in. An event can be one outcome or a group of outcomes.
Example
'Rolling an even number' is an event made of the outcomes 2, 4 and 6.
Outcomes are equally likely when each one has the same chance of happening.
An outcome that counts as the event you want happening.
Example
If the event is 'roll an even number', then 2, 4 and 6 are favourable outcomes.
An event that cannot happen. Its probability is 0.
An event that will definitely happen. Its probability is 1.
An event that is as likely to happen as not. Its probability is one half.
A coin, dice or spinner is fair when every outcome has an equal chance of happening.
A coin, dice or spinner is biased when some outcomes are more likely than others.
An activity with uncertain results, such as rolling a dice or flipping a coin.
The slip-ups students most often make with probability — and simple ways to avoid them.
Why it happens
The favourable and total outcomes are sometimes written the wrong way round, or counts are added instead of compared.
How to avoid it
Every probability must be between 0 and 1. If your answer is outside that range, check the calculation.
Why it happens
It is easy to lose track of which number is favourable and which is the total.
How to avoid it
Favourable outcomes always go on top, total outcomes on the bottom: .
Why it happens
The total is calculated quickly and one colour or group is left out.
How to avoid it
Add up every possible outcome carefully to find the total before writing the probability.
Why it happens
Without a fixed order it is hard to keep track of which outcomes have been written.
How to avoid it
Work systematically — use a table or a fixed order so every outcome is listed exactly once.
Why it happens
Unlikely sounds close to impossible, so the two are blurred together.
How to avoid it
An unlikely event has a probability greater than 0 — it still can happen, it just probably will not this time.
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A probability can never be less than 0 or more than 1. If you ever calculate a probability of or , something has gone wrong.
You can write a probability as a fraction, a decimal or a percentage — they all mean the same thing. An even chance is .
Think about some everyday events:
The word fair is important too. A fair coin or dice gives every outcome an equal chance. A biased one does not — perhaps it is weighted to favour a certain result.
Using the right word shows you understand roughly how likely an event is — and that is the first step before working out the exact number.
Challenge yourself
Some events overlap. On a dice, 'roll an even number' and 'roll a number bigger than 3' both include the outcomes 4 and 6. Can you list the outcomes that belong to both events?
Roll a fair dice. What is the probability of rolling an even number?
For the bag above — 3 red counters and 7 blue, 10 in total — the probability of picking a red counter is .
A useful check: the probabilities of all possible outcomes always add up to 1. Here .
Answer
Step 2
The favourable outcomes are the 6 yellow counters.
Step 3
Write the probability as favourable over total.
Step 4
Simplify the fraction by dividing the top and bottom by 2.
Answer
Step 2
Now take the second spinner colour, blue, and pair it with each coin result.
Step 3
Count the outcomes in the full list.
Answer
The outcomes are (Red, Heads), (Red, Tails), (Blue, Heads), (Blue, Tails) — there are outcomes.
Step 2
There are 4 outcomes in total, and each is equally likely.
Step 3
(b) Find the favourable outcomes — exactly one head. These are (H,T) and (T,H), so there are 2.
Step 4
Write the probability as favourable over total and simplify.
Answer
(a) (H,H), (H,T), (T,H), (T,T). (b) .