The idea, step by step
The condition changes the population
P(A|B) means the probability of A given that B has occurred. Keep only outcomes in B, then count those also in A. The denominator is the size or probability of B, not the whole original sample space.
For a fair die known to be greater than 3, only 4,5,6 remain. Two of these are even, so the conditional probability is 2/3 rather than the original 1/2.
Read the order of the events
P(girl|walk) and P(walk|girl) ask different questions. In the first, look only at walkers. In the second, look only at girls. The overlap count can be the same while the denominators differ.
Check that the condition is possible. A conditional probability given an event of probability zero is not found by dividing by zero. In finite counting questions, the restricted group must contain at least one outcome.
| Travel | Girl | Boy | Total |
|---|---|---|---|
| Cycle | 12 | 8 | 20 |
| Walk | 18 | 22 | 40 |
| Total | 30 | 30 | 60 |
A common trap
Dividing the overlap count by the whole total after a condition is given.
Instead: Divide by the count in the given group.
See the thinking
Restrict a die result
A fair die result is known to be odd. Find the probability it is prime.
- Keep {1,3,5}.
- The prime outcomes are 3 and 5.
A survey condition
Using the supplied table, find P(walk|girl).
| Travel | Girl | Boy | Total |
|---|---|---|---|
| Cycle | 12 | 8 | 20 |
| Walk | 18 | 22 | 40 |
| Total | 30 | 30 | 60 |
- There are 30 girls.
- 18 of those girls walk.
- 18/30=3/5.
Check your understanding
Using the table, find P(girl|walk).
Show answer
9/20.
18 of the 40 walkers are girls, so 18/40=9/20.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- A fair six-sided die result is known to exceed 3. Find P(even).2 marks
- A fair six-sided die result is known to be even. Find P(greater than 3).2 marks
- Find P(cycle|girl) from the 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(girl|cycle) from the 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 - Using the travel table, explain why P(cycle|girl) and P(girl|cycle) have different denominators.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 and B)=0.12 and P(B)=0.3. Find P(A|B).2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A fair six-sided die result is known to exceed 3. Find P(even).2 marks
Restrict to 4,5,6;4 and 6 are even.
2/3. - A fair six-sided die result is known to be even. Find P(greater than 3).2 marks
Restrict to 2,4,6;4 and 6 exceed 3.
2/3. - Find P(cycle|girl) from the 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 12/30=2/5.
2/5. - Find P(girl|cycle) from the 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 12/20=3/5.
3/5. - Using the travel table, explain why P(cycle|girl) and P(girl|cycle) have different denominators.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 Girls total 30, cyclists total 20.
The same overlap 12 is divided by different totals.
They condition on different groups. - P(A and B)=0.12 and P(B)=0.3. Find P(A|B).2 marks
0.12/0.3=0.4.
0.4.
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 ↗