Sample Space Diagrams
Grade 3What is a sample space?
A sample space is just a complete list of every possible outcome. When two things happen at the same time — two coins, a dice and a spinner — it helps to draw them in a grid called a sample space diagram.
One event goes along the top, the other down the side. Each cell in the grid is one possible combined outcome. Count the cells to find your total.
Worked example — two dice
Reading probabilities from the grid
Once the grid is drawn, every probability question becomes a counting exercise. Circle or highlight the cells where your event happens, count them, then divide by the total number of cells.
Try this one
A bag contains 3 red counters and 7 blue counters. A counter is taken out at random, its colour is recorded, and it is put back in the bag. A second counter is then taken out at random.
(a) Complete the tree diagram below by filling in the four missing probabilities.
First pick Second pick
Red (\( \frac{3}{10} \)) ——— Red ( __ )
Red ( __ ) <
Red (\( \frac{3}{10} \)) ——— Blue ( __ )
Blue ( __ ) <
Blue (\( \frac{7}{10} \)) ——— Red ( __ )
Blue ( __ ) <
Blue (\( \frac{7}{10} \)) ——— Blue ( __ )
[2]
(b) Work out the probability that both counters are the same colour. [3]
This uses tree diagrams with replacement — because the counter goes back, each pick is independent and the probabilities stay the same on every branch.
Ask yourself: 'How do I combine the probabilities along a branch to find the probability of two events both happening?'
For example, if P(A) = 1/4 and P(B) = 1/4 along a branch, the probability of that route is 1/4 × 1/4.
