Systematic lists, arrays, tables and trees | Year 9

01 / LEARN

The idea, step by step

Choose a useful representation

A list works for a small sample space. A two-way table is helpful when two stages each have a small number of options: a row identifies one stage and a column the other. Every cell is a complete outcome.

A tree is especially useful when later choices depend on earlier choices. Read from the start to one leaf for a full result. Internal branch points are not extra outcomes.

Follow a path or combine leaves

This tree represents a fair coin followed by an independent equal three-section spinner. Following H then 2 has probability $1 / 2 \times 1 / 3 = 1 / 6$ . There are six leaves.

An even spinner result means H2 or T2. These separate complete outcomes have total probability $1 / 6 + 1 / 6 = 1 / 3$ . Add only distinct favourable leaves.

A common trap

H2 and T2 are the same full outcome because both end in 2.

Instead: The first results differ, so they are different complete paths.

02 / WORKED EXAMPLES

See the thinking

Example 1

A letter-number code

A code uses A or B followed by 1, 2, 3 or 4. List all codes.

Fix A: A1, A2, A3, A4.
Fix B: B1, B2, B3, B4.

Eight codes: A1, A2, A3, A4, B1, B2, B3, B4.

Example 2

Two-dice sum

Two fair independent dice are rolled. Find P(sum 5).

Favourable ordered pairs are (1,4), (2,3), (3,2), (4,1).
There are $6 \times 6 = 36$ equally likely ordered pairs.

4 / 36 = 1 / 9

Check your understanding

How many leaves for 3 first-stage options and 5 second-stage options, with all pairings allowed?

Show answer

Each of three first branches has five continuations.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

List all ordered two-letter codes using X and Y, allowing repeats.
1 mark
A meal has 3 main choices and 2 drinks. How many complete meals can be formed?
2 marks
List the ordered outcomes with sum 4 when two dice are rolled.
2 marks
For two fair independent dice, find P(sum 4).
2 marks
Use the displayed tree to find P(H then 3).
2 marks
For a code A, B or C followed by 0 or 1, explain why A0, B1, C0 is incomplete.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

List all ordered two-letter codes using X and Y, allowing repeats.
1 mark
Hold the first letter fixed while changing the second.
XX, XY, YX, YY.
A meal has 3 main choices and 2 drinks. How many complete meals can be formed?
2 marks
A 3 by 2 array has six cells.
6
List the ordered outcomes with sum 4 when two dice are rolled.
2 marks
Try first-die results in order.
(1,3), (2,2), (3,1).
For two fair independent dice, find P(sum 4).
2 marks
Three of 36 outcomes are favourable.
$1 / 12$
Use the displayed tree to find P(H then 3).
2 marks
Multiply the branch probabilities: $1 / 2 \times 1 / 3$ .
$1 / 6$
For a code A, B or C followed by 0 or 1, explain why A0, B1, C0 is incomplete.
3 marks
Each letter needs both digits.
There should be six codes.
A1, B0 and C1 are missing.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 9 compound-event sample spaces, probabilities, experiments and simulations.

Official curriculum source ↗

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