Sample Space Diagram

Here we will learn how to use a sample space, including listing outcomes, creating sample space diagrams and finding probabilities.

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

What is a sample space diagram?

A sample space diagram is used to display all possible outcomes, this could be as a list or a table of values. Making a list of all possible outcomes is known as enumeration.

To create a sample space diagram we need to think about the possible outcomes of a situation.

For example,

Let’s say we were to flip a fair coin and at the same time roll a fair six-sided die.

The sample space for this situation could be written as a list of the possible combinations.

The sample space for more complicated situations is often shown in a table.

For example,

If flipping the coin and rolling the die were part of a game and the side the coin shows changed the score of the die, we could show this in a sample space diagram.

In this game, the coin showing heads adds 1 to the value on the die. The coin showing tails doubles the value on the die.

The sample space diagram shows all the possible outcomes for this game.

We can use sample space diagrams to find probabilities.

Jon plays the game and wants to know the probability of getting a score of more than 6.

We can see from the table that there are 4 scores more than 6.

There are a total of 12 possible outcomes from this game.

Using the theoretical probability formula,

The tables used for sample space diagrams can also be referred to as two-way tables.

Step-by-step guide: Two-way tables (coming soon)

What is a sample space?

How to use a sample space diagram

In order to find a probability using a sample space diagram:

1. Use information provided to decide whether to write a list or create a table to find all possible outcomes.
2. Systematically write the list or fill in the table by either listing outcomes or performing operations with values.
3. Use the information from the list or table to find any probabilities required.

Explain how to use a sample space

Related lessons on combined events

Sample space diagram 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:

• Combined events probability
• Product rule for counting
• Systematic listing strategies

Sample space diagram examples

Example 1: writing a list of possible outcomes

A fair coin is flipped and a 4 sided spinner numbered 1 to 4 is spun.

Write a list of all possible combinations of outcomes and find the probability of getting a tail and an even number.

1. Use information provided to decide whether to write a list or create a table to find all possible outcomes.

This question requires a list of outcomes. The coin has two outcomes – H and T, the spinner has 4 outcomes, 1, 2, 3 and 4. We can write them as pairs e.g. (H, 1).

We should have 4 options for each side of the coin, so 8 outcomes in total. It is best to list them in a systematic way to ensure we don’t miss any combinations.

2Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

3Use the information from the list or table to find any probabilities required.

There are two possible outcomes that have a Tail and an even number.

Example 2: creating a sample space table

Two fair spinners, one numbered 1, 2 and 3, the other 2, 5, and 7 are spun.

The scores on the spinners are added together. Find the probability that the total score is a prime number.

Use information provided to decide whether to write a list or create a table to find all possible outcomes.

The easiest way to organise the outcomes here is a table.

The spinners each have 3 outcomes so we should have 9 possible outcomes in total.

Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

Use the information from the list or table to find any probabilities required.

There are three possible outcomes that are prime.

\text{P(Total score is prime)}=\frac{3}{9}=\frac{1}{3}

Example 3: creating a sample space diagram using a rule

A fair coin is flipped and a fair six-sided die is rolled. If the coin shows heads, the score on the die is increased by 3. If the coin shows tails the score on the die is decreased by 1. Find the probability of getting a score less than 5.

Use information proved to decide whether to write a list or create a table to find all possible outcomes.

This question is best answered with a table. The coin has 2 outcomes and the die has 6 outcomes so we should have 12 possible outcomes in total.

Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

Use the information from the list or table to find any probabilities required.

There are six possible outcomes that are less than 5.

\text{P(Score less than 5)} = \frac{6}{12} = \frac{1}{2}.

Common misconceptions

• Missing out or repeating outcomes when listing

A common error is to miss out some outcomes or repeat one already listed. This can happen if we do not use a systematic approach to forming the list.

Creating a table to list the outcomes can help structure our list to ensure none are missed or repeated.

The next lessons are

• How to calculate probability
• Probability distribution
• Describing probability

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