The idea, step by step
Three different questions
An unconditional probability uses the full group. A joint probability asks for both characteristics within the full group. A conditional probability first restricts to a subgroup. Write the words before calculating so you do not accidentally answer the wrong question.
In the travel table, P(girl)=30/60, P(girl and walk)=18/60 and P(girl|walk)=18/40. All three calculations use real entries, but they answer different questions.
| Travel | Girl | Boy | Total |
|---|---|---|---|
| Cycle | 12 | 8 | 20 |
| Walk | 18 | 22 | 40 |
| Total | 30 | 30 | 60 |
Use proportions, not raw counts
To test independence, compare the conditional proportion with the unconditional proportion. Equal raw counts do not establish independence if the groups have different totals.
An association does not by itself prove causation. Survey tables can be affected by other factors, sampling choices and chance variation. Report a probability conclusion for the supplied model without inventing a causal explanation.
A common trap
Reading P(A|B) as P(A and B), or assuming conditional order can be swapped.
Instead: The condition changes the denominator. Write each event in words and identify the restricted group.
See the thinking
Unconditional versus conditional
Find P(girl) and P(girl|walk) in the travel table.
| Travel | Girl | Boy | Total |
|---|---|---|---|
| Cycle | 12 | 8 | 20 |
| Walk | 18 | 22 | 40 |
| Total | 30 | 30 | 60 |
- P(girl)=30/60=1/2.
- Among 40 walkers,18 are girls, so P(girl|walk)=9/20.
A table with independence
A class has 20 girls and 20 boys. Ten girls and ten boys cycle. Are gender and cycling independent in this table?
- P(cycle)=20/40=1/2.
- P(cycle|girl)=10/20=1/2 and likewise for boys.
Check your understanding
In the travel table, compare P(walk) and P(walk|boy).
Show answer
2/3 and 11/15.
Use 40/60 for all students and 22/30 within boys.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Find P(cycle) from the travel table.2 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 - Find P(boy and cycle).2 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 - Find P(boy|cycle).2 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 - Are being a boy and cycling independent in this table?3 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 - P(A)=0.25 and P(A|B)=0.25, with P(B)>0. What can you conclude?2 marks
- A survey links extra tutoring with higher scores. Explain why a conditional proportion alone does not prove tutoring caused the increase.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find P(cycle) from the travel table.2 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 20/60=1/3.
1/3. - Find P(boy and cycle).2 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 8/60=2/15.
2/15. - Find P(boy|cycle).2 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 8/20=2/5.
2/5. - Are being a boy and cycling independent in this table?3 marks
Invented survey of 60 students; every student is in exactly one travel category. Travel Girl Boy Total Cycle 12 8 20 Walk 18 22 40 Total 30 30 60 P(boy)=1/2, but P(boy|cycle)=2/5.
No. - P(A)=0.25 and P(A|B)=0.25, with P(B)>0. What can you conclude?2 marks
The condition does not change the probability of A.
A and B are independent. - A survey links extra tutoring with higher scores. Explain why a conditional proportion alone does not prove tutoring caused the increase.3 marks
Prior attainment, motivation and sampling can affect both groups.
A probability comparison alone does not isolate a cause.
Other differences may explain the association.
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 ↗