Conditional probability uses information already known about an outcome. That information changes the group you should count or the next branch of a tree. Drawing without replacement also changes both the number of objects and sometimes the favourable count. Multiply along a path for successive events, then add mutually exclusive paths that describe the requested outcome. For exactly one of each colour, include both possible orders. Write fractions before multiplying so changing denominators and the meaning of the condition remain visible.
Learning objectives
- Calculate dependent probabilities for draws without replacement.
- Use the condition to identify the correct denominator group.
Key ideas
Without replacement, reduce the total and the count of the colour removed.
Multiply probabilities along a path; add mutually exclusive paths for the requested outcome.
Conditioning restricts the sample space to the stated group.
In a two-way table, divide by the total of the group specified by the condition.
Worked examples
Example 1
A bag has four red and three blue counters. Draw two at random without replacement. Find the probability of exactly one of each colour.
Answer
Example 2
In the same bag, find the probability the second counter is red given the first is red.
Answer
Question styles to prepare for
Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.
AO1 · Methods
Complete a dependent tree or calculate a conditional probability.
AO2 · Reasoning
Explain how removing an item changes the next probability or denominator.
AO3 · Problem solving
Combine different branches or recover an unknown count from a probability.
When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.
Avoid this mistake
The second draw has a smaller total without replacement. Include both possible orders when finding exactly one of each colour.
Try it yourself
Attempt each question before opening its answer.
Check 1: In that bag, find P(second counter red given the first is red).
Answer: 3/6 = 1/2
Once a red is removed, 3 red counters remain out of 6.
Check 2: 12 of the 20 students who wear glasses are in a club. Find P(club given glasses).
Answer: 3/5
The condition restricts the group to the 20 glasses wearers, so use 12/20.
Explain it in your own words
- Explain why second-draw probabilities change without replacement.
- Write both orders for exactly one red and one blue.
- Explain which denominator to use when a condition is given.
Keep practising
Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes
Specification source
AQA Probability specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.