A coin is flipped once. How many possible outcomes are there?
Think: heads or tails.
Outcomes are H and T → 2 outcomes.
Listing Outcomes
Listing outcomes involves writing all possible results of an event. This ensures no possibilities are missed when calculating probability and helps form a complete set of outcomes for the experiment. It builds directly on basic probability.
Overview
Listing outcomes means writing down all the possible results of an event or experiment.
This is one of the most important basic probability skills.
Probability = \( \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}} \)
If outcomes are missed or repeated, the probability will be incorrect, so it is important to work systematically and stay organised.
What you should understand after this topic
- Understand what an outcome is
- List all possible outcomes without missing any
- Use a list of outcomes to calculate probabilities
- Work with one-step and two-step events
- Understand when order matters
Key Definitions
Outcome
A single possible result of an experiment.
Event
A result or group of results that we are interested in.
Favourable Outcome
An outcome that matches the event we want.
Total Outcomes
All the possible results altogether.
Systematic List
A list written in a clear order so nothing is missed or repeated.
Sample Space
The full set of all possible outcomes.
Key Rules
List in order
Be systematic so you do not miss any outcomes.
Do not repeat
Each different outcome should appear once only.
Check if order matters
\( HT \) and \( TH \) may count as different outcomes.
Count carefully
Use the full list to count total and favourable outcomes.
Quick Examples of Outcome Lists
One coin
\( H, T \)
One dice
\( 1, 2, 3, 4, 5, 6 \)
Two coins
\( HH, HT, TH, TT \)
Spinner
List every section once.
How to Solve
Step 1: Understand listing outcomes
Listing outcomes means writing every possible result of an experiment.
The full list of outcomes is called the sample space.
Exam tip: You must include every outcome exactly once.
Step 2: Understand the experiment
Decide what is happening in the question.
Exam thinking: More steps → more outcomes.
Single event
One coin or one dice.
Two events
Two coins or two dice.
Step 3: List outcomes systematically
Write outcomes in a clear pattern to avoid missing any.
Fix the first part
Keep the first result the same while changing the second.
Use a pattern
List all outcomes before moving on.
Avoid repetition
Each outcome should appear only once.
Step 4: Count outcomes
Count the total number of outcomes.
This is the denominator of the probability.
Step 5: Find favourable outcomes
Identify outcomes that match the event.
Count these to form the numerator.
Step 6: Order matters
In multi-step experiments, order can matter.
\( HH,\ HT,\ TH,\ TT \)
\( HT \neq TH \) because the order is different.
Exam tip: Treat each position separately.
Step 7: Two-event listing method
For two events, list systematically using pairs.
Start with the first result fixed (e.g. H),
List all possibilities for the second result,
Then move to the next first result.
This idea links directly to tree diagrams, which organise outcomes visually.
Step 8: Calculate probability
\( \text{Probability} = \dfrac{\text{Favourable outcomes}}{\text{Total outcomes}} \)
Simplify the fraction where possible.
See probability basics for more.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on listing outcomes and using them to find probabilities.
Edexcel
A coin is tossed once. List all the possible outcomes.
Edexcel
A fair dice is rolled once. List all the possible outcomes.
Edexcel
A coin is tossed twice. List all the possible outcomes.
Edexcel
A coin is tossed twice. Find the probability of getting exactly one head.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on systematic listing and equally likely outcomes.
AQA
Sam has one red card, one blue card and one green card. He chooses one card at random. List all the possible outcomes.
AQA
A spinner has four equal sections labelled A, B, C and D. The spinner is spun once. List all the possible outcomes.
AQA
A fair dice is rolled once. Find the probability of rolling a number greater than 4.
AQA
A fair dice is rolled once. Find the probability of rolling an even number.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising complete lists of outcomes and simple probability reasoning.
OCR
A coin is tossed and a fair dice is rolled. List all the possible outcomes.
OCR
A coin is tossed and a fair dice is rolled. Find the probability of getting a head and an even number.
OCR
A bag contains one red counter, one yellow counter and one blue counter. Two counters are chosen without replacement. List all the possible ordered outcomes.
OCR
Two counters are chosen without replacement from one red counter, one yellow counter and one blue counter. Find the probability that one of the counters is red.
Exam Checklist
Step 1
Work out exactly what experiment is happening.
Step 2
List every possible outcome in a clear order.
Step 3
Count the total number of outcomes carefully.
Step 4
Count how many outcomes match the event.
Most common exam mistakes
Missing outcomes
Not writing the full sample space.
Repeating outcomes
Counting the same result more than once.
Order mistake
Forgetting that order can matter in multi-step events.
Wrong count
Counting favourable outcomes incorrectly after listing them.
Common Mistakes
These are common mistakes students make when listing outcomes in GCSE Maths.
Missing outcomes
Incorrect
A student does not include all possible results.
Correct
List outcomes systematically to ensure none are missed. Use a table or organised list to cover every possibility.
Repeating outcomes
Incorrect
A student writes the same outcome more than once.
Correct
Each outcome should appear only once. Check your list carefully to avoid duplicates.
Ignoring order when it matters
Incorrect
A student treats outcomes like AB and BA as the same.
Correct
If order matters, AB and BA are different outcomes. Always check whether the situation depends on order.
Counting favourable outcomes incorrectly
Incorrect
A student identifies the correct outcomes but counts them wrongly.
Correct
After listing outcomes, count carefully. Double-check your totals to avoid simple errors.
Using an incomplete list
Incorrect
A student calculates probability from a partial list.
Correct
Probability depends on all possible outcomes. If the list is incomplete, the probability will be incorrect.
Try It Yourself
Practise listing outcomes systematically in probability problems.
Foundation
Foundation Practice
List all possible outcomes in simple situations.
List all outcomes when flipping a coin twice. Give as comma-separated values.
Combine first and second flip.
Possible outcomes: HH, HT, TH, TT.
A die is rolled. How many outcomes are possible?
Numbers 1 to 6.
6 possible outcomes.
List all outcomes when rolling a die once.
All numbers from 1 to 6.
Outcomes: 1, 2, 3, 4, 5, 6.
How many outcomes are there when flipping a coin and rolling a die?
Multiply outcomes.
2 × 6 = 12 outcomes.
How many outcomes are there when flipping two coins?
List them if unsure.
HH, HT, TH, TT → 4 outcomes.
A student lists HH, HT, TT when flipping two coins. What did they miss?
Order matters.
They missed TH.
How many outcomes are there when rolling a die twice?
6 × 6.
36 possible outcomes.
Which method helps ensure all outcomes are listed?
Be organised.
Systematic listing avoids missing outcomes.
How many outcomes are there when choosing between A, B and C?
Count the options.
There are 3 outcomes.
Higher
Higher Practice
List outcomes systematically and avoid duplicates in more complex situations.
Two dice are rolled. How many total outcomes are possible?
6 × 6.
Each die has 6 outcomes → 36 total.
List all outcomes when flipping a coin and rolling a die (start with H). Give as comma-separated values.
Combine H and T with numbers 1–6.
All combinations of H/T with 1–6.
How many outcomes are there when choosing one letter from A, B, C and one number from 1, 2?
3 × 2.
3 letters × 2 numbers = 6 outcomes.
List all outcomes when choosing from A, B and 1, 2. (Letter first).
Pair each letter with each number.
A1, A2, B1, B2.
A student lists outcomes for two dice but writes (1,2) and (2,1) only once. What is wrong?
Order matters.
(1,2) and (2,1) are different outcomes.
How many outcomes are there when choosing from 4 options and 3 options?
Multiply 4 × 3.
12 total outcomes.
Which list is systematic?
Look for clear pattern.
This list follows a structured pattern.
How many outcomes are there when flipping 3 coins?
2 × 2 × 2.
8 outcomes.
A student forgets to list some outcomes. What is the main problem?
Missing outcomes changes totals.
Missing outcomes leads to incorrect probabilities.
How many outcomes are there when rolling 2 dice and flipping a coin?
6 × 6 × 2.
72 total outcomes.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What are outcomes?
Possible results.
Why list systematically?
To avoid missing cases.
What helps?
Tables or diagrams.