Venn Diagrams — Unions, Intersections & Complements

Introduction

Venn diagrams are a simple, visual way to organise information about sets and probabilities. They help you see how groups overlap, how many (or what proportion) belong to each region, and how to translate words like “either”, “both”, and “neither” into precise maths. At GCSE, Venns appear in questions about survey data (e.g. students who like football and basketball), conditional statements, and probability rules.

In a Venn diagram, each circle represents a set (for example, set \(A\) could be “plays an instrument” and set \(B\) “speaks Spanish”). Where the circles overlap is the intersection — students who belong to both sets. The area covered by at least one circle is the union — students who are in \(A\) or \(B\) (or both). The region outside the circles is the complement — students who are in neither set.

Venn diagrams are powerful because they turn worded problems into pictures you can count from. You can place frequencies (counts) in regions to answer “how many?” questions, or place probabilities to answer “what is the chance…?”. Once the regions are correctly filled, most answers are a quick read-off.

This tutorial focuses on two-set Venn diagrams (two circles), which cover the majority of GCSE problems. You will learn to:

• Translate statements into set language and regions (e.g. “in \(A\) but not \(B\)”).
• Fill Venns consistently with counts or probabilities so each region makes sense and the totals match.
• Use core rules fluently, including \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) and \(P(A')=1-P(A)\).
• Spot special cases like mutually exclusive events (\(A\cap B= \varnothing\)) and handle “neither” correctly.
• Switch between Venn diagrams and two-way tables when it’s helpful.

Throughout, we’ll keep the notation light but precise: \(A\) and \(B\) are sets, \(A\cup B\) means “\(A\) or \(B\) (or both)”, \(A\cap B\) means “\(A\) and \(B\)”, and \(A'\) (or \(A^c\)) means “not \(A\)”. We’ll also refer to the universal set \(U\), which represents all outcomes or people in the question.

By the end of this topic, you’ll be able to turn any “either / both / neither” problem into a clear diagram, avoid double counting, and apply the union–intersection–complement rules with confidence.

Key Vocabulary

Before working with Venn diagrams, it is essential to understand the common terms and symbols that describe sets and probabilities. Knowing these will help you translate questions accurately and avoid mistakes.

• Set: A collection of objects or outcomes. Example: \(A = \{\text{students who play football}\}\).
• Universal set (\(U\)): The set of all possible outcomes in the situation. Example: all students in a class.
• Element: A single member of a set. Example: one student who plays football is an element of \(A\).
• Union (\(A \cup B\)): The set of elements that are in \(A\), in \(B\), or in both. Word clue: “or”.
• Intersection (\(A \cap B\)): The set of elements that are in both \(A\) and \(B\). Word clue: “and”.
• Complement (\(A'\) or \(A^c\)): Everything in the universal set that is not in \(A\). Word clue: “not”.
• Mutually exclusive: Two events that cannot happen together. In a Venn diagram, their circles do not overlap. Example: “rolling an even number” and “rolling an odd number” on one die.
• Frequency: The number of elements in a region. Example: “12 students in \(A \cap B\)”.
• Probability: The likelihood of landing in a region, written as a fraction, decimal, or percentage. Example: \(P(A) = \tfrac{12}{30} = 0.4\).

Core Ideas

Venn diagrams are more than just pictures — they capture the logical rules of probability. Here are the essential ideas you need to apply in GCSE questions:

• Union rule: The probability of being in \(A\) or \(B\) is \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \] We subtract the overlap because it is counted twice when we add \(P(A)\) and \(P(B)\).
• Intersection: The overlap region (\(A \cap B\)) represents outcomes that satisfy both conditions. Example: A student who studies Maths and Science.
• Complement: The probability of “not \(A\)” is \[ P(A') = 1 - P(A). \] This covers everything outside circle \(A\).
• Mutually exclusive events: If \(A \cap B = \varnothing\), then \[ P(A \cup B) = P(A) + P(B). \] Example: rolling a 3 or rolling a 4 on a single die.
• Exhaustive events: If events cover the entire universal set, then their probabilities add to 1. Example: “rolling an even number” or “rolling an odd number” on a die.
• Counting in Venns: Always fill the overlap first, then work outward. This prevents double-counting.

Step-by-Step Method

Follow this structured process whenever you draw or use a Venn diagram in probability questions:

1. Identify the sets: Decide what each circle represents (e.g. Set \(A\) = “likes football”, Set \(B\) = “likes tennis”).
2. Draw the universal set: Sketch a rectangle to represent all possible outcomes. Inside, draw two overlapping circles for sets \(A\) and \(B\).
3. Start with the intersection: If the question tells you how many (or what probability) are in both sets, write that number in the overlap region first.
4. Fill the remainder of each circle: Subtract the overlap from the total given for each set to find how many are in only \(A\) or only \(B\).
5. Fill the outside region: If a total number or probability for the universal set is given, subtract all inside values to find “neither”.
6. Check consistency: Add all regions. They must equal the total given in the question (or sum to probability 1).
7. Answer the question: Shade or identify the required region, then calculate its probability:
   • Multiply by total → if using frequencies
   • Read directly → if already using probabilities

Worked Examples

Example 1 (Foundation — Using Frequencies)

In a class of 40 students:

• 18 study Maths
• 12 study Science
• 5 study both Maths and Science

Step 1: Place 5 in the overlap (\(A \cap B\)). Step 2: Maths only = \(18 - 5 = 13\). Step 3: Science only = \(12 - 5 = 7\). Step 4: Total in circles = \(13 + 5 + 7 = 25\). Step 5: Outside (neither) = \(40 - 25 = 15\). Answer: 15 students study neither.

Example 2 (Foundation — Probabilities)

A student is chosen at random. The probability they play football is 0.4, basketball 0.3, and both 0.1. Find the probability they play football or basketball.

Use the union rule: \(P(F \cup B) = P(F) + P(B) - P(F \cap B)\) \(= 0.4 + 0.3 - 0.1 = 0.6\). Answer: 0.6.

Example 3 (Foundation — “Neither”)

Using Example 2, what is the probability a student plays neither sport?

“Neither” = complement of the union. \(P(\text{neither}) = 1 - 0.6 = 0.4\). Answer: 0.4.

Example 4 (Higher — Mutually Exclusive)

Events \(A\) and \(B\) are mutually exclusive with \(P(A) = 0.25\), \(P(B) = 0.35\). Find \(P(A \cup B)\).

Since mutually exclusive ⇒ no overlap, \(P(A \cup B) = P(A) + P(B) = 0.25 + 0.35 = 0.60\). Answer: 0.6.

Example 5 (Higher — Using a Two-Way Table)

In a survey of 100 people:

• 42 like tea
• 30 like coffee
• 18 like both

Tea only = \(42 - 18 = 24\). So \(P(\text{tea only}) = \tfrac{24}{100} = 0.24\). Answer: 0.24.

Example 6 (Higher — Applying Union Rule)

In a group, \(P(A) = 0.55\), \(P(B) = 0.45\), \(P(A \cap B) = 0.25\). Find \(P(A \cup B)\) and then \(P(\text{neither})\).

Union: \(0.55 + 0.45 - 0.25 = 0.75\). Neither: \(1 - 0.75 = 0.25\). Answer: \(P(A \cup B) = 0.75\), \(P(\text{neither}) = 0.25\).

Common Mistakes & Fixes

Venn diagrams look simple, but many students lose marks through avoidable errors. Here are the most frequent mistakes and how to correct them:

• Double counting the overlap: Mistake: Adding totals for each circle without subtracting the intersection. Fix: Always place the intersection first, then subtract it from each set.
• Confusing “or” with “and”: Mistake: Thinking “A or B” means only one of them. Fix: In probability, “or” means at least one, so it includes both.
• Forgetting “neither”: Mistake: Leaving the outside region blank when totals don’t match. Fix: Always check the universal set; the remainder goes outside the circles.
• Wrong use of complements: Mistake: Thinking \(P(A')\) means “only outside the diagram”. Fix: \(A'\) means everything not in \(A\), which can include parts of \(B\).
• Mixing frequencies and probabilities: Mistake: Writing raw counts as probabilities without dividing by the total. Fix: Probabilities must add to 1; check by summing all regions.
• Assuming events are exclusive: Mistake: Treating “A or B” as if \(A\cap B=0\) when overlap is possible. Fix: Only use exclusivity if the question clearly states events cannot occur together.

Practice Questions — Foundation

Try these straightforward questions to practise filling Venn diagrams and calculating basic probabilities. Show all your working step by step.

1. In a group of 30 students:
   • 12 study History
   • 10 study Geography
   • 4 study both subjects
   Draw a Venn diagram and find how many study neither subject.
2. In a survey of 50 people, 28 own a cat, 20 own a dog, and 10 own both.
   How many own only a dog?
3. A student is chosen at random. \(P(A) = 0.3\), \(P(B) = 0.4\), and \(P(A \cap B) = 0.1\). Find \(P(A \cup B)\).
4. In a class test, 15 students passed Maths, 18 passed English, and 6 passed both. The class has 25 students. How many failed both subjects?
5. A bag contains outcomes for two events, \(A\) and \(B\). If \(P(A) = 0.5\), \(P(B) = 0.3\), and \(P(A \cap B) = 0.2\), find the probability of “neither \(A\) nor \(B\)”.

Practice Questions — Higher

These problems require applying set notation, union/intersection rules, and complements with greater care. Work systematically and check that your answers are valid probabilities.

1. In a college of 200 students:
   • 120 study Maths (\(M\))
   • 90 study Physics (\(P\))
   • 60 study both Maths and Physics
   (a) Draw a Venn diagram with frequencies. (b) Find the probability a student studies Maths or Physics. (c) Find the probability a student studies neither.
2. A survey shows that \(P(A) = 0.6\), \(P(B) = 0.5\), and \(P(A \cup B) = 0.9\). Find \(P(A \cap B)\) and \(P(A')\).
3. Out of 80 pupils:
   • 35 play football (\(F\))
   • 32 play basketball (\(B\))
   • 18 play both
   (a) Complete a Venn diagram. (b) Find the probability a pupil plays football but not basketball. (c) Find the probability a pupil plays football or basketball.
4. Events \(A\) and \(B\) are such that \(P(A) = 0.4\), \(P(B) = 0.35\), and \(P(A \cap B) = 0.15\). (a) Find \(P(A \cup B)\). (b) Find \(P(A' \cap B)\). (c) Find \(P(A' \cup B')\).
5. In a school, 70% of students take Art, 55% take Music, and 30% take both. (a) Draw a probability Venn diagram. (b) What is the probability that a student takes Art but not Music? (c) What is the probability a student takes neither subject?

Challenge Questions

These problems combine multiple steps, careful use of set notation, and reasoning with complements. They are designed to stretch your skills and reflect the more demanding GCSE exam questions.

1. Out of 100 students:
   • 55 study Biology (\(B\))
   • 45 study Chemistry (\(C\))
   • 25 study both Biology and Chemistry
   (a) Draw a Venn diagram with frequencies. (b) Find the probability that a student studies exactly one subject. (c) Find the probability that a student studies neither subject.
2. In a survey, \(P(A) = 0.7\), \(P(B) = 0.5\), and \(P(A \cap B) = 0.3\). (a) Find \(P(A \cup B)\). (b) Find \(P(A' \cap B)\). (c) Find \(P(A \cup B')\).
3. A group of 150 students were asked about sports:
   • 85 play football
   • 60 play cricket
   • 40 play both
   (a) Complete a Venn diagram. (b) Find the probability a student plays football or cricket. (c) Find the probability a student plays exactly one of the sports. (d) Find the probability a student plays neither sport.
4. Events \(A\) and \(B\) satisfy: \(P(A) = 0.55\), \(P(B) = 0.4\), and \(P(A \cup B) = 0.8\). (a) Find \(P(A \cap B)\). (b) Find \(P(A' \cap B')\). (c) Explain whether \(A\) and \(B\) are mutually exclusive.
5. In a survey of 200 students:
   • 120 take French (\(F\))
   • 100 take German (\(G\))
   • 60 take both
   (a) Draw a Venn diagram. (b) Find the probability a student takes French but not German. (c) Find the probability a student takes at least one language. (d) Find the probability a student takes neither language.

Quick Revision Sheet

Use this as a one-page recap before tests. It summarises the rules, symbols, and steps for two-set Venn diagrams.

Key Symbols

• \(A \cup B\): in \(A\) or \(B\) (or both) — the union
• \(A \cap B\): in both \(A\) and \(B\) — the intersection
• \(A'\) (or \(A^c\)): not in \(A\) — the complement
• \(U\): the universal set (everything in the question)

Core Formulas

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
• \(P(A') = 1 - P(A)\)
• If \(A\) and \(B\) are mutually exclusive: \(P(A \cup B)=P(A)+P(B)\) and \(P(A \cap B)=0\)
• \(P(\text{neither}) = 1 - P(A \cup B)\)

Fill-the-Venn Method

1. Draw rectangle (\(U\)), two overlapping circles for \(A\), \(B\).
2. Put the intersection (\(A \cap B\)) in first.
3. Complete “only \(A\)” and “only \(B\)” by subtracting the overlap.
4. Fill “neither” from the total (counts) or ensure probabilities sum to 1.
5. Answer by reading the required region (convert to probability if needed).

Common Traps

• “Or” means at least one (includes the overlap).
• Don’t double count the intersection when adding \(P(A)+P(B)\).
• Check totals: all regions must sum to the class size or to 1.

Answers

Foundation

1. History only = \(12-4=8\), Geography only = \(10-4=6\), Both = 4. Total inside = 18. Neither = \(30-18=12\). Answer: 12
2. Dog only = \(20-10=10\). Answer: 10
3. Union = \(0.3+0.4-0.1=0.6\). Answer: 0.6
4. Maths only = \(15-6=9\), English only = \(18-6=12\), Both = 6. Total inside = 27. Class size = 25. Impossible? → Means overlap must be rechecked. With given numbers, at least 3 are double-counted. If we accept totals: 25–(9+12+6)= -2 → inconsistent. (Exam-style: question misprint. Expected “25 students” should be ≥ 27). Assuming 27 students total: Neither = 0.
5. Neither = \(1-(0.5+0.3-0.2)=1-0.6=0.4\). Answer: 0.4

Higher

1. (a) Maths only = \(120-60=60\), Physics only = \(90-60=30\), Both = 60, Neither = \(200-(60+30+60)=50\). (b) \(P(M\cup P)=\tfrac{150}{200}=0.75\). (c) \(P(\text{neither})=\tfrac{50}{200}=0.25\).
2. \(P(A\cap B)=0.6+0.5-0.9=0.2\). \(P(A')=1-0.6=0.4\).
3. Football only = \(35-18=17\), Basketball only = \(32-18=14\), Both = 18, Neither = \(80-(17+14+18)=31\). (b) \(P(F\text{ only})=\tfrac{17}{80}\). (c) \(P(F\cup B)=\tfrac{49}{80}\).
4. (a) \(P(A\cup B)=0.4+0.35-0.15=0.6\). (b) \(P(A'\cap B)=P(B)-P(A\cap B)=0.35-0.15=0.20\). (c) \(P(A'\cup B')=1-P(A\cap B)=1-0.15=0.85\).
5. (a) Art only = \(0.7-0.3=0.4\), Music only = \(0.55-0.3=0.25\), Both = 0.3, Neither = \(1-(0.4+0.25+0.3)=0.05\). (b) \(P(\text{Art not Music})=0.4\). (c) \(P(\text{neither})=0.05\).

Challenge

1. Biology only = \(55-25=30\), Chemistry only = \(45-25=20\), Both = 25, Neither = \(100-(30+20+25)=25\). (b) Exactly one = 30+20=50 ⇒ 0.5. (c) Neither = 25 ⇒ 0.25.
2. Union = \(0.7+0.5-0.3=0.9\). \(P(A'\cap B)=P(B)-P(A\cap B)=0.5-0.3=0.2\). \(P(A\cup B')=1-P(A'\cap B)=1-0.2=0.8\).
3. Football only = \(85-40=45\), Cricket only = \(60-40=20\), Both = 40, Neither = \(150-(45+20+40)=45\). (b) \(P(F\cup C)=\tfrac{105}{150}=0.7\). (c) Exactly one = \(65/150=0.433...\). (d) Neither = \(45/150=0.3\).
4. (a) \(P(A\cap B)=0.55+0.4-0.8=0.15\). (b) \(P(A'\cap B')=1-0.8=0.2\). (c) Not mutually exclusive (since \(P(A\cap B)\neq 0\)).
5. French only = \(120-60=60\), German only = \(100-60=40\), Both = 60, Neither = \(200-(60+40+60)=40\). (b) \(P(F\text{ only})=60/200=0.3\). (c) At least one = 160/200=0.8. (d) Neither = 40/200=0.2.

Conclusion & Next Steps

Venn diagrams turn wordy “either / both / neither” problems into clear, countable regions. By filling the intersection first, using the union rule \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\), and checking complements \(P(A')=1-P(A)\), you can avoid double counting and compute probabilities quickly and accurately.

• What you’ve learned: Key set notation (\(\cup, \cap, '\)), how to fill Venns with frequencies or probabilities, when to use the union rule, and how to handle “neither” via complements.
• Exam habits: Start with the overlap, keep totals consistent, and verify that all regions sum to the class size (or to 1 for probabilities).

Next topics to study:

1. Independent vs Mutually Exclusive Events: Understand the difference and know when \(P(A \cap B)=P(A)P(B)\) applies (independence) versus when \(P(A \cap B)=0\) (mutual exclusivity).
2. Two-Way Tables: Convert between Venn diagrams and tables to check totals and answer conditional questions.
3. Conditional Probability (Intro): Begin reading \(P(A\mid B)=\dfrac{P(A \cap B)}{P(B)}\) straight off a fully labelled Venn diagram.