Sample Space Diagrams

In a Nutshell

A sample space diagram lists every possible outcome of two combined events, making it easy to count favourable outcomes and find probabilities.

When two events happen together — for example, rolling two dice — the number of combined outcomes can be large. A sample space diagram is a two-way table that organises every possible combination.

For two six-sided dice, there are 6×6=366 \times 6 = 36 outcomes in total. To find the probability of any result, count how many cells in the table match and divide by 36.

For instance, to find the probability that the sum of two dice is 7, look for cells containing 7 in the table. There are 6 such cells: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)(1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

P(sum=7)=636=16P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}

Sample space diagram A two-way grid showing every possible combined outcome from two events. Tap a cell to highlight it and see the outcome. Die 1 Die 2

Watch it work

Question: Two fair dice are rolled and the scores are added. Find the probability that the total is 10.

Have a go

Q1. Two dice are rolled. Find the probability the sum is 2.

Q2. Two dice are rolled and the scores are multiplied. Find the probability the product is 12.

Q3. Two dice are rolled and scores are subtracted (larger minus smaller). Find the probability the difference is 0.

Q4. How many of the 36 outcomes give a sum greater than 9?