Understand the idea.
Tables and complete paths
A table is useful for combining two short outcome lists. On a tree, follow branches from the start to an endpoint to read a complete outcome. A first-stage branch alone does not yet describe both stages of the experiment.
Probabilities on branches leaving one point total 1 because they exhaust what can happen next. Multiply the probabilities along a complete path using the probabilities shown for its stages. Add separate endpoint probabilities when the desired event can happen through different paths.
Venn regions are disjoint
A Venn diagram splits the group into disjoint regions: first activity only, both activities, second activity only and neither. The overlap belongs to both activity sets but represents each person once in the whole-group total.
In the supplied diagram, cycling contains students and swimming contains . At least one activity contains . Adding the 4 outside both circles gives the full group of 20 without double-counting anyone.
See the method in action.
Worked example 1
Find the complete outcomes ending in blue on the supplied tree.
Worked solution
- 1
Trace every endpoint ending in blue.
Either coin result can be followed by blue, so there are two complete favourable paths.
- 2
Multiply along each path.
The tree gives a blue probability of one quarter after either first-stage result.
- 3
Add the mutually exclusive paths.
A single trial cannot end in both HB and TB, so these path probabilities add.
HB or TB; probability 1/4.
Worked example 2
Find how many do neither activity in the supplied Venn diagram.
Worked solution
- 1
Count the disjoint regions inside the circles.
Cycling-only, both and swimming-only together contain everyone doing at least one activity.
- 2
Subtract from the whole group.
Neither is the region outside both circles but inside the group rectangle.
- 3
Check the complete partition.
The four non-overlapping regions must account for all twenty students.
4 students.
Try it yourself.
Check your understanding
How many students cycle in the supplied Venn diagram?
Show answer & reasoning
12.
The cycling circle includes both cycling-only and the overlap. Therefore students cycle, including those who also swim.
Keep practising
6 worksheet questions · 11 marks
Read the questions online
- How many swim but do not cycle in the diagram?1 mark
- Find in the diagram.2 marks
- List all complete outcomes on the tree.2 marks
- Using the tree, find .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 in the diagram.2 marks
Add once each.
. - 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 .3 marks
; .
Add disjoint outcomes:.
. - A two-stage table has 3 rows and 4 columns of outcomes. Count its cells.1 mark
.
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 ↗