Conditional probability is one of the most challenging Higher-tier topics in GCSE Maths. It deals with finding the probability of an event given that another event has already occurred. Venn diagrams and two-way tables are the main tools used to organise the information and calculate these probabilities. This guide explains the formula, the set notation, and how to apply both methods confidently.
What Is Conditional Probability?
Conditional probability is the probability of event A happening given that event B has already happened. It is written as P(A|B) and read as "the probability of A given B".
The key idea is that once you know B has occurred, the sample space shrinks. You are no longer considering all possible outcomes — only those where B is true. Within that reduced group, you count how many also satisfy A.
Venn diagrams are particularly useful for conditional probability because they show overlapping groups clearly. The intersection A ∩ B contains outcomes in both A and B, while A ∪ B contains outcomes in either or both. The region outside both circles represents outcomes in neither A nor B. Two-way tables achieve the same organisation using rows and columns.
Key Formulas
Step-by-Step Method
- Organise the data into a Venn diagram or two-way table.
- Identify what you need: P(A|B) means "given B has happened, what is the chance of A?"
- Find P(A ∩ B) — the number (or probability) in both A and B.
- Find P(B) — the total number (or probability) in B.
- Divide: P(A|B) = P(A ∩ B) ÷ P(B).
Worked Example 1 — Foundation Level
Question: 50 students were asked whether they study French (F) or Spanish (S). 18 study both, 28 study French, 30 study Spanish. A student is chosen at random from those who study Spanish. Find the probability they also study French.
Working: P(F|S) means: given the student studies Spanish, what is the probability they study French? Number studying both (F ∩ S) = 18. Number studying Spanish = 30. P(F|S) = 18/30 = 3/5.
Answer: P(F|S) = 3/5 or 0.6.
Worked Example 2 — Higher Level
Question: In a class of 40 students, 22 play football (F), 15 play tennis (T), and 8 play both. A student who plays football is chosen at random. Find the probability they do not play tennis.
Working: Football only = 22 − 8 = 14 students. Given the student plays football, total in F = 22. P(not T | F) = 14/22 = 7/11.
Answer: P(does not play tennis | plays football) = 7/11.
Worked Example 3 — Exam Style
Question: The two-way table shows data about 100 adults.
| Drives | Does not drive | Total | |
|---|---|---|---|
| Male | 35 | 15 | 50 |
| Female | 30 | 20 | 50 |
| Total | 65 | 35 | 100 |
Find P(Male | Drives).
Working: P(Male | Drives) = number who are male AND drive ÷ total who drive. P(Male | Drives) = 35/65 = 7/13.
Answer: P(Male | Drives) = 7/13.
Common Mistakes
- Confusing P(A|B) with P(B|A). P(A given B) is not the same as P(B given A). Always check which event is the condition (the "given" part) — that becomes your denominator.
- Using the total instead of the condition group. For P(A|B), the denominator is the number in B (not the overall total). The condition restricts the sample space.
- Forgetting to subtract the intersection. When filling in a Venn diagram, the intersection must be placed first, then subtracted from each circle total to find the "only" regions.
Exam Tips
- Always draw and label a Venn diagram or two-way table, even if the question does not ask for one. It organises the information and earns method marks.
- Write out the conditional probability formula before substituting. Examiners award marks for stating P(A|B) = P(A ∩ B) / P(B).
- Check that all regions of your Venn diagram add up to the total. This catches errors before they affect your answer.
Practice Questions
Q1 (Foundation): 60 people were surveyed. 25 like tea (T), 35 like coffee (C), and 10 like both. Draw a Venn diagram and find P(T ∩ C).
Q2 (Foundation): Using the data from Q1, find P(T | C).
Q3 (Higher): In a group of 80 students, 45 study biology (B), 30 study chemistry (C), and 20 study both. Find P(C | B) and P(B | C).
Practise conditional probability questions with instant feedback — completely free on GCSEMathsAI.
Related Topics
Summary
- Conditional probability P(A|B) is the probability of A given B has occurred.
- Use the formula P(A|B) = P(A ∩ B) ÷ P(B). The denominator is always the condition group.
- Venn diagrams and two-way tables are the best tools for organising conditional probability data.
- P(A|B) and P(B|A) are different — always check which event is the condition.
- Fill in Venn diagrams starting with the intersection, then the "only" regions, then the "neither" region.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.
Probability investigations and games from Cambridge.
University of Cambridge · Free · Open AccessTree diagrams, Venn diagrams, and conditional probability.
Corbett Maths · Free · Open AccessMIT introduction to probability theory.
Massachusetts Institute of Technology · Free · Open Access