Sample Space

The list of potential outcomes from a probability question is known as the sample space. For example, the sample space for rolling a 6-sided dice would be {1, 2, 3, 4, 5, 6}.

Let’s look at an example:

Example

A class of girls and boys groups together to form teams of 5 children. Write down all the possible numbers of girls and boys in each team.

Solution:

Write down all the possible combinations to make up a team of 5. The order doesn’t matter in this case so we can summarise it as:

• 5 boys

• 4 boys, 1 girl

• 3 boys, 2 girls

• 2 boys, 3 girls

• 1 boy, 4 girls

• 5 girls

Tip: When writing outcomes as a list, make sure to work logically through all the solutions to make sure that you don’t miss any options.

Sample Space diagrams

Sometimes, there are a lot of potential outcomes. For example, if two dice are rolled and the numbers are added together, what are the potential outcomes this time?

We would need to work out each combination individually: 1+1, 1+2, etc. This would take a long time to write as a list and makes it easy to miss solutions.

Here is where a sample space diagram comes in. It’s like a two-way table to help organise the outcomes.

Below is a sample space diagram for the example above. The numbers in the grid are the sums of all the possible combinations of rolls from both dice.

We can use this grid to solve probability questions. It is clear to see that there are 36 potential outcomes .

Example Questions

Question 1:

What is the probability of the total being an even number?

Solution

Circle all the even numbers in the table.

There are 18. Therefore, the probability would be

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Question 2:

What is the probability of the total being a multiple of 4?

Solution

Circle all the multiples of 4 in the table.

There are 9. Therefore, the probability would be

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Question 3:

What is the probability of the outcome being less than 5?

Solution

Circle all the numbers less than 5. However, don’t include 5!

There are 6. Therefore, the probability would be