AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 22

2 marks · Medium difficulty · Short Answer

Use a Venn diagram and the given probability P(A) = 1/2 to find the missing value x.

Practise this question

Question

A Venn diagram showing universal set xi with two intersecting circles labelled A and B. The region in A only has 30, the intersection has 20, the region in B only has x, and the region outside both circles has 6. Underneath the diagram it states P(A) = 1/2 and asks to work out the value of x.

Mark scheme

Show the mark scheme Mark scheme for question 22 showing M1 for 2 × (30 + 20) or 100, or 20 + 30 - 6, or 50 - 6, or an equivalent equation such as x + 6 = 50. Followed by A1 for the final answer 44, with additional guidance that M1 may be awarded for correct work even if no answer or an incorrect answer is given.

How to answer it

Venn Diagrams: Finding an Unknown Frequency from Probability

📋 What this question tests
  • Venn Diagram Interpretation: Recognizing that the frequency of set A includes both the independent region and the intersection: n(A) = 30 + 20 .
  • Theoretical Probability: Using the formula P(A) = n(A) ÷ Total to deduce the total frequency.
  • Forming and Solving Equations: Summing all 4 disjoint regions of a Venn diagram to equal the universal set total.

Question 22 (2 Marks)

Work out the value of x given P(A) = 1/2

📐 Step-by-Step Calculations

Method 1: Finding the Total Universal Set (ξ)

  1. Find the total number of items in Set A:
    n(A) = 30 + 20 = 50
  2. Find the grand total:
    Since P(A) = 1/2, the total must be double the size of A:
    Total = 2 × 50 = 100 [M1]
  3. Add all regions and solve for x:
    30 + 20 + x + 6 = 100
    56 + x = 100
    x = 100 − 56 = 44 [A1]

Method 2: Using the Complement (Outside A)

  • If P(A) = 1/2, then the items not in A must also equal 1/2 of the total.
  • Items inside A = 30 + 20 = 50
  • Items outside circle A = x + 6
  • Therefore: x + 6 = 50 [M1]
  • x = 50 − 6 = 44 [A1]

💡 Key Knowledge

  • Complete Circle: Set A consists of everything inside circle A: 30 (only A) + 20 (both A and B) .
  • Total Elements: Every item in the Venn diagram belongs to one of four mutually exclusive regions:
    • Only A ( 30 )
    • Both A and B ( 20 )
    • Only B ( x )
    • Neither A nor B ( 6 )
  • Probability Relation: P(A) = n(A) ÷ n(ξ) . If this equals 1/2, exactly half of all items are inside circle A, and the other half are outside.

✅ Final Answer

x = 44

Mark Scheme Breakdown:
• M1: For 2 × (30 + 20) or 100 OR 50 − 6 OR setting up x + 6 = 50 (or equivalent equation).
• A1: For final answer 44 (also credited if clearly written in the region on the diagram).

🧠 Exam Technique & Examiner Tips

  • Always check by substitution: Add up all regions with your answer to verify:
    30 + 20 + 44 + 6 = 100
    P(A) = 50 / 100 = 1/2 . It matches perfectly!
  • Show your working clearly: Even if you make an arithmetic error when subtracting, showing 100 or x + 6 = 50 secures the method mark (M1).

❌ Common Errors to Avoid

  • Forgetting the intersection: Assuming set A contains only 30 elements instead of 30 + 20 = 50 .
  • Ignoring the outside region: Forgetting to include the 6 items outside the circles when calculating the total.
  • Halving instead of doubling: Calculating 50 ÷ 2 = 25 rather than realizing that 50 represents half of the universal set.

Topics

Probability · Algebra · 3.5 Probability · 3.2.3 Solving equations and inequalities

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.