GCSE Maths Statistics and Probability - How to Solve: Venn Diagrams and Set Notation

How to Solve: Venn Diagrams and Set Notation

GCSE / A-Level (Physics, Chemistry, Biology) – Complete Guide

Introduction

"Mastering Venn diagrams and set notation lets you crack 3–5 mark questions in every science exam—questions that separate a Grade 4 from a Grade 7. In biology, you’ll classify organisms; in chemistry, you’ll sort elements by properties; in physics, you’ll group forces or circuits. Lose these marks, and you lose a whole grade. Get them right, and you’re already ahead."

WHAT YOU NEED TO KNOW FIRST

1. Basic set symbols: ∪ (union), ∩ (intersection), ⊆ (subset), ∅ (empty set).
2. Number of elements in a set: Written as n(A) for set A.
3. Simple probability: P(A) = n(A) / n(total).

KEY TERMS & FORMULAS

STEP-BY-STEP METHOD

Step 1: Read the question carefully

• Underline all sets mentioned (e.g., A, B, ξ).
• Circle key words: "only", "both", "neither", "all".

Step 2: Draw the Venn diagram framework

• Draw two overlapping circles (for two sets) or three (for three sets).
• Label each circle with the set name (A, B, C).
• Draw a rectangle around them and label it ξ (universal set).

Step 3: Fill in the intersection first

• Start with the overlap (A ∩ B).
• Write the number of elements only if given. If not, leave it as x.

Step 4: Fill in the "only" regions

• Subtract the intersection from each set’s total to find the "only A" and "only B" regions.
• Only A = n(A) – n(A ∩ B)
• Only B = n(B) – n(A ∩ B)

Step 5: Fill in "neither" (outside the circles)

• Neither = n(ξ) – n(A ∪ B)
• Use the inclusion-exclusion formula if needed:
• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Step 6: Answer the question

• Count elements: Add up the relevant regions.
• Write in set notation: Use ∪, ∩, or ’ as needed.
• Calculate probability: P(A) = n(A) / n(ξ).

WORKED EXAMPLES

Example 1 – Basic (Two Sets)

Question: In a class of 30 students: - 18 study Biology (B). - 12 study Chemistry (C). - 5 study both. How many study only Biology?

Step-by-Step Solution: 1. Read: Sets = B, C; ξ = 30 students. 2. Draw: Two overlapping circles (B and C). 3. Fill intersection: B ∩ C = 5. 4. Fill "only" regions:
- Only B = n(B) – n(B ∩ C) = 18 – 5 = 13.
- Only C = n(C) – n(B ∩ C) = 12 – 5 = 7. 5. Neither: n(ξ) – n(B ∪ C) = 30 – (18 + 12 – 5) = 30 – 25 = 5. 6. Answer: 13 students study only Biology.

What we did and why: We isolated the "only Biology" region by subtracting the overlap from the total Biology students. This avoids double-counting the students who take both subjects.

Example 2 – Medium (Three Sets)

Question: A survey of 50 people: - 25 like tea (T). - 20 like coffee (C). - 15 like juice (J). - 8 like tea and coffee. - 5 like coffee and juice. - 3 like tea and juice. - 2 like all three. How many like only tea?

Step-by-Step Solution: 1. Read: Sets = T, C, J; ξ = 50. 2. Draw: Three overlapping circles. 3. Fill intersection (all three): T ∩ C ∩ J = 2. 4. Fill pairwise intersections:
- T ∩ C only = 8 – 2 = 6.
- C ∩ J only = 5 – 2 = 3.
- T ∩ J only = 3 – 2 = 1. 5. Fill "only" regions:
- Only T = n(T) – (T ∩ C only + T ∩ J only + all three) = 25 – (6 + 1 + 2) = 16.
- Only C = n(C) – (T ∩ C only + C ∩ J only + all three) = 20 – (6 + 3 + 2) = 9.
- Only J = n(J) – (C ∩ J only + T ∩ J only + all three) = 15 – (3 + 1 + 2) = 9. 6. Neither: n(ξ) – (Only T + Only C + Only J + T ∩ C only + C ∩ J only + T ∩ J only + all three) = 50 – (16 + 9 + 9 + 6 + 3 + 1 + 2) = 4. 7. Answer: 16 people like only tea.

What we did and why: We worked from the center outwards (all three → pairwise → only) to avoid missing overlaps. This ensures no region is double-counted or left empty.

Example 3 – Exam-Style (Disguised Question)

Question: A lab tests 80 samples for two properties: X and Y. - 45 samples have property X. - 30 samples have property Y. - 10 samples have neither property. What is the probability a randomly selected sample has both properties?

Step-by-Step Solution: 1. Read: Sets = X, Y; ξ = 80; n(X) = 45; n(Y) = 30; Neither = 10. 2. Find n(X ∪ Y): n(ξ) – Neither = 80 – 10 = 70. 3. Use inclusion-exclusion:
- n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y).
- 70 = 45 + 30 – n(X ∩ Y).
- n(X ∩ Y) = 45 + 30 – 70 = 5. 4. Calculate probability:
- P(X ∩ Y) = n(X ∩ Y) / n(ξ) = 5 / 80 = 1/16. 5. Answer: The probability is 1/16.

What we did and why: We used the inclusion-exclusion principle to find the overlap, then converted it to a probability. This is a common exam trick—hiding set notation in a probability question.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Listen up—this is your 60-second Venn diagram survival guide. First, draw the circles and label them. Always start with the overlap (A ∩ B). Then fill the ‘only’ regions by subtracting the overlap from each set’s total. For ‘neither’, subtract the union from the universal set. If you see three sets, work from the center outwards—all three first, then pairwise, then ‘only’. Watch out for hidden ‘neither’ or probability questions—they’re traps! Finally, double-check your arithmetic—examiners love to test if you can add and subtract correctly. You’ve got this!"