Sample Space Diagrams

What Is a Sample Space?

The sample space of a probability experiment is the complete list of all possible outcomes.

• For a coin toss: Sample space = $\{H, T\}$
• For a six-sided dice roll: Sample space = $\{1, 2, 3, 4, 5, 6\}$

When you combine more than one event (e.g. two dice rolls), the total number of possible outcomes increases quickly. To help manage this, we use sample space diagrams (also called possibility diagrams).

 
What Is a Sample Space Diagram?

A sample space diagram is a grid or table used to clearly display all possible outcomes when combining two or more events.

Common Uses:

• Rolling two dice
• Flipping two or three coins
• Drawing two counters from a bag

These diagrams help you count and spot outcomes easily.

 

Example: Rolling Two Dice and Adding Their Scores

We’ll draw a grid where each row and column represents the result of each dice.

 

	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

 

Each cell shows the sum of two dice. There are 36 outcomes in total (6 rows × 6 columns).

 

How to Use a Sample Space Diagram to Calculate Probability

To calculate a probability:

$P(A) = \frac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}$

You must be sure that each outcome in the diagram is equally likely (i.e. fair dice, unbiased coins, etc.).

 

Example

Question: Two fair six-sided dice are rolled.

(a) Find the probability that the total is a multiple of 4.

Step 1: List all outcomes where the sum is a multiple of 4

From the sample space:

• 4: (1,3), (2,2), (3,1)
• 8: (2,6), (3,5), (4,4), (5,3), (6,2)
• 12: (6,6)

Total of 9 outcomes.

Total possible outcomes = 36

$P(\text{multiple of 4}) = \frac{9}{36} = \frac{1}{4}$

 

(b) Given that the total is a multiple of 4, find the probability that at least one of the dice shows a 6.

From the 9 favourable outcomes:

• (2,6), (6,2), (6,6) include a 6.

So, 3 outcomes contain at least one six.

$P(\text{at least one 6 | multiple of 4}) = \frac{3}{9} = \frac{1}{3}$