Understand the idea.
Choose a useful representation
Choose a representation that makes missing outcomes easy to spot. A systematic list fixes the first choice while varying the second. In a two-way table or array, each row-column pairing records one complete two-stage outcome.
A tree records stages in sequence and can show how later choices depend on earlier results. Follow branches all the way to a leaf to read a complete outcome; a point halfway through the tree is only a partial result.
Follow a path or combine leaves
In the displayed tree, H followed by 2 is one full path. Its probability is . Multiplication follows the stages of one path, using the probability written on each branch.
An even spinner result can occur through H2 or T2. These are different, non-overlapping outcomes, so add their probabilities: . Keep the first-stage result when identifying leaves, even if the event mentions only the spinner.
See the method in action.
A letter-number code
A code uses A or B followed by 1, 2, 3 or 4. List all codes.
Worked solution
- 1
List all codes beginning with A.
Holding the first character fixed lets each allowed number appear exactly once.
- 2
List all codes beginning with B.
B is the only other first character, and the same four numbers remain available.
- 3
Check that the list is complete.
The two groups contain four distinct codes each, covering every permitted letter-number pairing.
Eight codes: A1, A2, A3, A4, B1, B2, B3, B4.
Two-dice sum
Two fair independent dice are rolled. Find P(sum 5).
Worked solution
- 1
Count all ordered dice outcomes.
Each first-die value has six possible second-die values; fairness and independence make these pairs equally likely.
- 2
List the ordered pairs whose sum is 5.
Trying first-die values in order gives exactly four valid second-die values.
- 3
Calculate the probability from the two counts.
The four listed pairs are distinct favourable outcomes among the 36 equally likely pairs.
.
Try it yourself.
Check your understanding
How many leaves for 3 first-stage options and 5 second-stage options, with all pairings allowed?
Show answer & reasoning
15
Each of the three first-stage choices has five continuations. With all pairings allowed, there are leaves, each representing one complete outcome.
Keep practising
6 worksheet questions · 12 marks
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 .2 marks
- Use the displayed tree to find .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.
- List the ordered outcomes with sum 4 when two dice are rolled.2 marks
Try first-die results in order.
, , . - For two fair independent dice, find .2 marks
Three of 36 outcomes are favourable.
- Use the displayed tree to find .2 marks
Multiply the branch probabilities: .
- 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.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics
Year 9 compound-event sample spaces, probabilities, experiments and simulations.
Official curriculum source ↗