Conditional Probability & Venn Diagrams –

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, t

§Key definitions

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".

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.

Answer:

P(F|S) = 3/5 or 0.6.

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).

§Formulas to memorise

P(A|B) = P(A ∩ B) ÷ P(B)

P(A ∩ B) = number in both A and B ÷ total

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

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".

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

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.

⚠ 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.

MMXXVI specification · AQA · Edexcel · OCRgcsemathsai.co.uk/formulas/conditional-probability-venn-diagrams

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