Conditional Probability
Understand conditional probability for Queensland Year 11 Mathematical Methods (QCAA). Conditional probability measures the chance of one event given that another has already happened, working within a reduced sample space.
You will learn to read conditionals from two-way tables, Venn diagrams and tree diagrams, apply the multiplication rule linking joint and conditional probabilities, and find the complement of a conditional — an essential stepping stone into QCAA statistics.
Every question with a fully worked solution.
- Conditional Probability - Video - Conditional probability Watch
Theory
In Year 11 Mathematical Methods (QCAA, Unit 1 Topic 5), a conditional probability \(P(A\mid B)\) is the probability of \(A\) given that \(B\) has happened. It uses the reduced sample space of \(B\): \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\). This page reads conditionals from tables, Venn diagrams and trees.
A conditional probability \(P(A\mid B)\) is read ‘the probability of \(A\) given \(B\)’. Knowing \(B\) has occurred restricts attention to the outcomes in \(B\) — the reduced sample space.
We divide the overlap by the condition: \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\). With counts this is \(\dfrac{n(A\cap B)}{n(B)}\) — the favourable count over the size of \(B\), not the whole sample space.
Rearranging gives the multiplication rule \(P(A\cap B)=P(A\mid B)\,P(B)\). The complement of a conditional keeps the same condition: \(P(A'\mid B)=1-P(A\mid B)\).
| Glasses | No glasses | Total | |
|---|---|---|---|
| Left-handed | 12 | 18 | 30 |
| Right-handed | 28 | 42 | 70 |
| Total | 40 | 60 | 100 |
The conditional probability formula:
With counts (reduced sample space):
The multiplication rule (rearranged):
How to find a conditional probability
- Spot the condition: the event after ‘given’ is \(B\); it becomes the new denominator.
- Find the overlap: \(P(A\cap B)\) or the count \(n(A\cap B)\) inside \(B\).
- Divide: \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{n(A\cap B)}{n(B)}\).
- Rearrange if needed: use \(P(A\cap B)=P(A\mid B)P(B)\), or the complement \(P(A'\mid B)=1-P(A\mid B)\).
Apply the conditional formula:
| \(P(A\mid B)\) | \(=\) | \(\dfrac{P(A\cap B)}{P(B)}\) |
| \(=\) | \(\dfrac{0.24}{0.4}\) |
Divide:
| \(=\) | \(0.6\) |
\(P(A\mid B)=0.6\).
| Owns a pet | No pet | Total | |
|---|---|---|---|
| Male | 60 | 40 | 100 |
| Female | 70 | 30 | 100 |
| Total | 130 | 70 | 200 |
‘Given female’ restricts to the female row: \(n(\text{female})=100\).
| \(n(\text{female}\cap\text{pet})\) | \(=\) | \(70\) |
| \(n(\text{female})\) | \(=\) | \(100\) |
Divide within the reduced sample space:
| \(P(\text{pet}\mid\text{female})\) | \(=\) | \(\dfrac{70}{100}\) |
| \(=\) | \(\dfrac{7}{10}\) |
\(P(\text{pet}\mid\text{female})=\dfrac{7}{10}\).
Given the first is red, one red is removed: \(3\) red remain of \(9\).
| \(P(\text{2nd red}\mid\text{1st red})\) | \(=\) | \(\dfrac{3}{9}\) |
| \(=\) | \(\dfrac{1}{3}\) |
\(P(\text{2nd red}\mid\text{1st red})=\dfrac{1}{3}\).
Rearrange for the overlap:
| \(P(A\cap B)\) | \(=\) | \(P(A\mid B)\,P(B)\) |
| \(=\) | \(0.7\times 0.5\) | |
| \(=\) | \(0.35\) |
Complement of the conditional (same condition \(B\)):
| \(P(A'\mid B)\) | \(=\) | \(1-P(A\mid B)\) |
| \(=\) | \(1-0.7\) | |
| \(=\) | \(0.3\) |
\(P(A\cap B)=0.35\) and \(P(A'\mid B)=0.3\).
Common pitfalls
Frequently asked questions
What is conditional probability?
\(P(A\mid B)\) is the probability of \(A\) given that \(B\) has occurred, found by \(\dfrac{P(A\cap B)}{P(B)}\).
What does ‘reduced sample space’ mean?
Once \(B\) is given, only outcomes in \(B\) are possible, so \(B\) becomes the new total (denominator).
How do you read a conditional probability from a two-way table?
Restrict to the row or column for the condition, then divide the overlap cell by that total, e.g. \(\dfrac{n(A\cap B)}{n(B)}\).
Is P(A given B) the same as P(B given A)?
No — they usually differ because they divide by different totals, \(P(B)\) versus \(P(A)\).
What is the multiplication rule?
\(P(A\cap B)=P(A\mid B)\,P(B)\), found by rearranging the conditional formula.
What is the complement of a conditional probability?
\(P(A'\mid B)=1-P(A\mid B)\); the condition \(B\) is unchanged.