1. A three sided spinner labelled $ A,B,C $ and a coin labelled $ H $ and $ T $ are used to create outcomes. The spinner is spun and the coin flipped. What is the correct list of possible outcomes?

$ AH, AT, BH, BT, CH, CT, HA $

6. A coin is flipped three times and the result is recorded. Make a list of the possible outcomes.

$ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT $

$ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT, HTH, THT $

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Product rule for counting How to calculate probability

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GCSE Maths Probability Combined Events

Systematic Listing

Systematic Listing Strategies

Here we will learn about systematic listing strategies, including calculating probabilities.

There are also counting strategies worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are systematic listing strategies?

Systematic listing strategies are ways of writing outcomes from an event in an organised way with none of the possibilities missed out or repeated. Writing all possible items in a set is known as enumeration.

To do this we need to use a method which makes listing items easier. A good method to use is to fix certain values, making only small changes between each item in the list.

For example,

A meal deal consists of a drink and a bag of crisps.

The drinks available are orange juice or apple juice.

The crisp flavours are bacon, cheese or plain.

To start listing all the possible combinations, we can begin by shortening the names of each flavour of drink and crisps.

Drinks – $katex is not defined$ and $katex is not defined$

Crisps – $katex is not defined$ and $katex is not defined$

Start by fixing the choice of drink to $katex is not defined$ and then change the crisp flavour.

katex is not defined

Now change the drink flavour and repeat the crisp flavours in the same order as before.

katex is not defined

This gives us $katex is not defined$ combinations in total.

This can be checked using the product rule for counting.

The were $katex is not defined$ flavours of drink and $katex is not defined$ flavours of crisps, so the total number of possible outcomes is $katex is not defined$

Step-by-step guide: Product rule for counting

What are systematic listing strategies?

Once we have listed the possible combinations, these outcomes can be used to calculate the probability of events occurring.

For example,

A $katex is not defined$ sided spinner labelled $katex is not defined$ is spun and a double sided counter labelled $katex is not defined$ and $katex is not defined$ is flipped.

List the possible combinations that could be obtained and find the probability that the sum of the combination is $katex is not defined$

Using systematic listing, we can see that the possible combinations are,

$katex is not defined$ sum to $katex is not defined$

There are $katex is not defined$ possible combinations and two of them sum to $katex is not defined$

Therefore, the probability of obtaining a sum of $katex is not defined$ is $katex is not defined$

How to use systematic listing strategies

In order to use systematic listing strategies:

Use a single letter or number for each item to be selected.
Determine the number of items to be selected.
Fix a first choice item. Follow a pattern for selecting second, third choices etc, fixing those where appropriate to ensure no options missed or unnecessarily duplicated. Repeat with a new first choice item.

Explain how to use systematic listing strategies

Probability of events worksheet (includes systematic listing strategies)

Get your free systematic listing strategies worksheet of 20+ probability of events questions and answers. Includes reasoning and applied questions.

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Probability of events worksheet (includes systematic listing strategies)

Get your free systematic listing strategies worksheet of 20+ probability of events questions and answers. Includes reasoning and applied questions.

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Related lessons on combined events

Systematic listing strategies is part of our series of lessons to support revision on combined events. You may find it helpful to start with the main combined events lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Systematic listing examples

Example 1: listing outcomes selecting from two sets

A coin is flipped and a spinner numbered $katex is not defined$ and $katex is not defined$ is spun.

Write a list of the possible outcomes.

Use a single letter or number for each item to be selected.

Use $katex is not defined$ and $katex is not defined$ for the sides of the coin and numbers $katex is not defined$ for the spinner.

2Determine the number of items to be selected.

One item from each is to be selected.

3Fix a first choice item. Follow a pattern for selecting second, third choices etc, fixing those where appropriate to ensure no options missed or unnecessarily duplicated. Repeat with a new first choice item.

Start by fixing the coin and change the spinner number.

katex is not defined

Now change the side of the coin and repeat the order of the spinner.

katex is not defined

There are $katex is not defined$ outcomes in total.

Example 2: listing outcomes selecting from three sets

A meal offer at a restaurant gives you a choice of main, side and drink.

Main	Side	Drink
Burger	Garlic Bread	Juice
Fish and Chips	Salad	Lemonade
Pasta		Cola

List the possible combinations of main, side and drink.

Use a single letter or number for each item to be selected.

Use $katex is not defined$ and $katex is not defined$ for the mains.

Use $katex is not defined$ and $katex is not defined$ for the sides.

Use $katex is not defined$ and $katex is not defined$ for the drinks.

Determine the number of items to be selected.

One item from each is to be selected.

Fix a first choice item. Follow a pattern for selecting second, third choices etc, fixing those where appropriate to ensure no options missed or unnecessarily duplicated. Repeat with a new first choice item.

Start by fixing the main, then the side and change the Drink, then change the side and repeat the pattern of the drinks.

$katex is not defined$

Now change the main, repeating the process for the sides and drinks:

$katex is not defined$

There are $katex is not defined$ outcomes in total.

Example 3: listing outcomes to give a set total

A car park charges $katex is not defined$ for a day of parking. The ticket machine only accepts $katex is not defined$ and $katex is not defined$ coins. List all the possible combinations of $katex is not defined$ and $katex is not defined$ coins to make a total of $katex is not defined$ The order of the coins does not matter.

Use a single letter or number for each item to be selected.

Use $katex is not defined$ and $katex is not defined$

Determine the number of items to be selected.

There will be different amounts of each but the sum of each combination must be $katex is not defined$

As the order of the coins does not matter, $katex is not defined$ is the same as $katex is not defined$

Start with using a 50p coin, introduce a 20p and then without a 20p.

$katex is not defined$

Now list combinations without a $katex is not defined$ coin, but using $katex is not defined$ coins, reducing the number of $katex is not defined$ coins each time.

$katex is not defined$

Last combination is just $katex is not defined$ coins

$katex is not defined$

There are $katex is not defined$ combinations in total.

Example 4: listing outcomes to find a number of items from a diagram.

The diagram shows a shape made from five points $katex is not defined$ and $katex is not defined$ is a straight line.

Find the number of triangles in the diagram.

Use a single letter or number for each item to be selected.

To list the triangles we will use three of the letters $katex is not defined$ and $katex is not defined$

Determine the number of items to be selected.

Each triangle must include the point $katex is not defined$ as the others are on a straight line. So the combinations will be $katex is not defined$ and two other letters.

The order of the letter does not matter, so triangle $katex is not defined$ is the same triangle as $katex is not defined$

Start by fixing A and B and change the third letter.

$katex is not defined$

Now change second letter in the combination and repeat the process

$katex is not defined$ notice that we do not write $katex is not defined$ since this is the same as $katex is not defined$ .

There are $katex is not defined$ triangles in total.

Common misconceptions

Lists are created randomly with no pattern

Common errors are made when students forget to follow a pattern or do not “fix” items in their combinations. It is important to use a systematic approach to ensure all combinations are listed. Small changes for each combination help with this.

Students do not consider if order matters

When listing outcomes it is important to think about the order of the combinations and whether it matters. For example, if flipping two coins, does the situation need $katex is not defined$ to be different to $katex is not defined$

Practice systematic listing strategies questions

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Select one letter from each of the spinner and coin and make pairs.

katex is not defined

katex is not defined

katex is not defined

katex is not defined

$katex is not defined$ and $katex is not defined$ are both humanities options so cannot both be selected.

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Fix $katex is not defined$ and $katex is not defined$ as they are the first in each category and then change the creative option.

katex is not defined

katex is not defined

katex is not defined

katex is not defined

There are $katex is not defined$ small squares, $katex is not defined$ which are $katex is not defined$ by $katex is not defined$ and $katex is not defined$ which are $katex is not defined$ by $katex is not defined$

katex is not defined

katex is not defined

katex is not defined

katex is not defined

The possible ways of getting $katex is not defined$ or more points are $katex is not defined$ wins, $katex is not defined$ wins and a draw, $katex is not defined$ wins and a loss or a win and $katex is not defined$ draws.

Using systematic listing we get,

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

There are $katex is not defined$ possible outcomes if the order of the combinations matters. A single coin is being flipped three times so this suggests the order will matter.

Systematic listing strategies GCSE questions

1. A coin is flipped and a $katex is not defined$ sided spinner labelled $katex is not defined$ to $katex is not defined$ is spun.

List the possible outcomes. The first one is done for you.

katex is not defined

(2 marks)

Show answer

All combinations with $katex is not defined$ included.

(1)

Complete list $katex is not defined$ .

(1)

2. A football team has four games remaining of the season and needs to achieve at least $katex is not defined$ points to avoid relegation.

A win gives $katex is not defined$ points, a draw gives $katex is not defined$ point, and a loss gives $katex is not defined$ points.

Using $katex is not defined$ for a win, $katex is not defined$ for a draw and $katex is not defined$ for a loss, list the possible combinations that will help the football team achieve at least $katex is not defined$ points.

(3 marks)

Show answer

A systematic approach to listing seen.

(1)

At least $katex is not defined$ combinations, which include some with L.

(1)

All $katex is not defined$ combination listed.

(1)

$katex is not defined$ .

3. A free meal for kids at a cafe offers a choice of sandwich, drink and fruit.

Sandwich	Drink	Fruit
Cheese (C) Ham (H) Egg (E)	Water (W) Smoothie (S)	Banana (B) Apple (A)

List all of the possible meal combinations.

(3 marks)

Show answer

Systematic approach seen.

(1)

All combinations with one sandwich seen.

(1)

All $katex is not defined$ combination given.

(1)

katex is not defined

Learning checklist

You have now learned how to:

Apply systematic listing strategies, including use of the product rule for counting

The next lessons are

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