The idea, step by step
Tables and complete paths
A table suits two short lists of outcomes. A tree keeps stages and their branches visible. Follow a path to an endpoint to obtain a complete outcome. Probabilities of all branches leaving the same point must total 1; a branch by itself describes only part of the experiment.
Venn regions are disjoint
Separate first-only, both, second-only and neither. The overlap belongs to both sets but is counted once in the group total. In the diagram, cycling contains 7+5=12, swimming 5+4=9, either activity 7+5+4=16, and all four regions total 20.
A common trap
Add both set totals without considering overlap.
Instead: That counts the intersection twice; instead add disjoint regions or subtract the overlap once.
See the thinking
Worked example 1
Find the complete outcomes ending in blue on the supplied tree.
- The endpoints ending B are HB and TB.
- Each has probability 1/2×1/4=1/8.
- Their total is 1/4.
Worked example 2
Find how many do neither activity in the supplied Venn diagram.
- Inside the circles: 7+5+4=16.
- Whole group:20.
- 20−16=4.
Check your understanding
How many students cycle in the supplied Venn diagram?
Show answer
12.
Add 7 cycling-only and 5 in both.
Your turn
6 questions · 11 marks · Work at your own pace.
Read the questions online
- How many swim but do not cycle in the diagram?1 mark
- Find P(cycling or swimming) in the diagram.2 marks
- List all complete outcomes on the tree.2 marks
- Using the tree, find P(HR or TB).3 marks
- A two-stage table has 3 rows and 4 columns of outcomes. Count its cells.1 mark
- A branch point has only two alternatives with probabilities 0.4 and 0.7. Explain the error.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- How many swim but do not cycle in the diagram?1 mark
Read the swimming-only region.
4 students. - Find P(cycling or swimming) in the diagram.2 marks
Add 7+5+4 once each.
16/20=4/5. - List all complete outcomes on the tree.2 marks
Each coin result is followed by R or B.
HR,HB,TR,TB. - Using the tree, find P(HR or TB).3 marks
P(HR)=3/8; P(TB)=1/8.
Add disjoint outcomes:4/8=1/2.
1/2. - A two-stage table has 3 rows and 4 columns of outcomes. Count its cells.1 mark
3×4=12.
12. - A branch point has only two alternatives with probabilities 0.4 and 0.7. Explain the error.2 marks
All alternatives leaving one point must sum to 1.
They total 1.1, not 1.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.
Official curriculum source ↗