The idea, step by step
Tables preserve the group counts
A two-way table classifies the same people by two characteristics. An inside cell counts an overlap; margins count complete groups. For a conditional probability, the relevant margin becomes the denominator.
In the club table,8 students do both activities. P(music|drama)=8/18, whereas P(drama|music)=8/20. Read the condition first and then choose the denominator.
| Music membership | Drama | Not drama | Total |
|---|---|---|---|
| Music | 8 | 12 | 20 |
| Not music | 10 | 30 | 40 |
| Total | 18 | 42 | 60 |
Branches are conditional on the earlier route
At each node of a probability tree, outgoing probabilities sum to 1. A second-stage label is the probability given the first-stage result. Multiply labels along a complete path to find its joint probability.
If several mutually exclusive paths satisfy the question, add their probabilities. For exactly one red counter, add the RB and BR paths; do not also include RR.
A common trap
Adding branch probabilities along a single two-stage path.
Instead: Multiply along a path; add distinct disjoint paths that satisfy the event.
See the thinking
A conditional club count
Using the table, find P(music|drama).
| Music membership | Drama | Not drama | Total |
|---|---|---|---|
| Music | 8 | 12 | 20 |
| Not music | 10 | 30 | 40 |
| Total | 18 | 42 | 60 |
- The condition limits us to 18 drama students.
- 8 of them also do music.
Exactly one red
Two counters are drawn without replacement from 3 red and 2 blue. Find P(exactly one red).
- RB has probability(3/5)(2/4)=3/10.
- BR has probability(2/5)(3/4)=3/10.
- These paths are disjoint, so add.
Check your understanding
In the tree, find P(two blue).
Show answer
1/10.
Multiply(2/5)(1/4)=1/10.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Using the club table, find P(drama|music).2 marks
Invented membership data for 60 students. Music membership Drama Not drama Total Music 8 12 20 Not music 10 30 40 Total 18 42 60 - Find P(not drama|not music).2 marks
Invented membership data for 60 students. Music membership Drama Not drama Total Music 8 12 20 Not music 10 30 40 Total 18 42 60 - Find P(music and drama).2 marks
Invented membership data for 60 students. Music membership Drama Not drama Total Music 8 12 20 Not music 10 30 40 Total 18 42 60 - Use the tree to find P(two red).2 marks
- Use the tree to find P(at least one red).3 marks
- Explain how to check that a completed four-leaf tree is consistent.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Using the club table, find P(drama|music).2 marks
Invented membership data for 60 students. Music membership Drama Not drama Total Music 8 12 20 Not music 10 30 40 Total 18 42 60 8/20=2/5.
2/5. - Find P(not drama|not music).2 marks
Invented membership data for 60 students. Music membership Drama Not drama Total Music 8 12 20 Not music 10 30 40 Total 18 42 60 30/40=3/4.
3/4. - Find P(music and drama).2 marks
Invented membership data for 60 students. Music membership Drama Not drama Total Music 8 12 20 Not music 10 30 40 Total 18 42 60 8 of all 60 do both;8/60=2/15.
2/15. - Use the tree to find P(two red).2 marks
(3/5)(2/4)=3/10.
3/10. - Use the tree to find P(at least one red).3 marks
The only excluded outcome is BB with probability 1/10.
1−1/10=9/10.
9/10. - Explain how to check that a completed four-leaf tree is consistent.3 marks
For this tree the leaf probabilities are 3/10,3/10,3/10,1/10.
Their sum is 1.
Every split totals 1, and all leaf probabilities total 1.
Curriculum reference & lesson scope
Australian Curriculum v 9 / SA Curriculum
Compound events; systematic counting and the multiplication principle; factorials and simple arrangements; conditional language and restricted sample spaces; conditional probabilities from tables/trees; replacement and dependence; conditional versus unconditional probabilities; designing and interpreting simulations.
Official curriculum source ↗