AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 17

3 marks · Medium difficulty · Multi-step Problem

Calculate how many times more likely a player is to spin four heads than alternating heads and tails given the ratio of heads to tails is 5 : 2.

Practise this question

Question

Exam question showing text: 'When a biased coin is spun, P(Heads) : P(Tails) = 5 : 2. In a game, the player spins the coin four times. The player can win in two ways.' Two boxes are shown: '1st way to win: Spin 4 heads' and '2nd way to win: Spin a Head, then a Tail, then a Head, then a Tail.' Below the boxes, the question asks: 'How many times more likely is a player to win the 1st way than the 2nd way?' with lines for working and an answer line, worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme for question 17 worth 3 marks. M1 is given for finding (5/7)^4 (625/2401) or (5/7)^2 * (2/7)^2 (100/2401). M1dep is given for dividing (5/7)^4 by [(5/7)^2 * (2/7)^2], or 625/2401 divided by 100/2401. A1 is given for 6.25 or 6 1/4 or 625/100. Additional guidance states working with decimals, percentages, or cancelling to 5^4 / (5^2 * 2^2) is acceptable.

How to answer it

Biased Coin: Comparing Combined Probabilities

📋 What this question tests

This question assesses your ability to:

  • Convert a given ratio into individual probabilities: P(event) = parts / total parts .
  • Calculate probabilities of successive independent events by multiplying probabilities along a path.
  • Interpret the phrase "how many times more likely" as a division of two probabilities ( P₁ ÷ P₂ ).
  • Work accurately with fractions or decimals under exam conditions.

Question 17 Walkthrough (3 Marks)

Comparing Outcomes from Four Successive Spins

📐 Step-by-Step Calculation

Step 1: Find individual probabilities from the ratio

Ratio of Heads to Tails = 5 : 2
Total parts = 5 + 2 = 7
• P(Head) = 5/7
• P(Tail) = 2/7

Step 2: Calculate P(1st way to win) — Four Heads (HHHH)

P(HHHH) = (5/7) × (5/7) × (5/7) × (5/7) = (5/7)⁴ = 625 / 2401

Step 3: Calculate P(2nd way to win) — Head, Tail, Head, Tail (HTHT)

P(HTHT) = (5/7) × (2/7) × (5/7) × (2/7) = (5/7)² × (2/7)² = 100 / 2401

Step 4: Determine how many times more likely

P(1st way) ÷ P(2nd way) = (625 / 2401) ÷ (100 / 2401)

= 625 / 100 = 6.25 (or 25/4 or 6 ¼)

✅ Correct Answer & Marks

Final Answer: 6.25 or 6 ¼ or 25/4 or 625/100

Mark Breakdown:
• M1: Correct probability expression for either outcome, e.g. (5/7)⁴ [= 625/2401] or (5/7)² × (2/7)² [= 100/2401].
• M1dep: Fully correct division method, e.g. (5/7)⁴ ÷ ((5/7)² × (2/7)²) or 625/2401 ÷ 100/2401 .
• A1: Accurate final answer: 6.25 or equivalent fraction.

💡 Key Knowledge & Pro Shortcut

  • The "Times More Likely" Rule: When comparing event A to event B, compute P(A) ÷ P(B) , not a difference ( P(A) - P(B) ).
  • Denominator Shortcut: Both sequences have exactly 4 spins, so their denominators are both 7⁴ = 2401. They cancel out completely:
    Ratio = 5⁴ ÷ (5² × 2²) = 5² ÷ 2² = 25 ÷ 4 = 6.25
    Examiner note: Writing 5⁴ ÷ (5² × 2²) or 625 ÷ 100 scores both M1 M1 instantly!

🧠 Exam Technique & Examiner Insight

  • Keep Fractions Where Possible: Converting 5/7 to decimals gives recurring numbers (≈ 0.714...). Working in fractions keeps values exact and prevents rounding errors.
  • Parentheses Matter: If using a calculator for division, make sure brackets enclose the denominator: (5/7)^4 / ((5/7)^2 * (2/7)^2) . Missing brackets without recovery will lose method marks.
  • Percentages: Working with rounded percentages (e.g. 71% and 29%) is accepted if carried out correctly, but fraction arithmetic is faster and much safer.

❌ Common Errors to Avoid

  • Subtracting instead of dividing: Calculating P(1st) - P(2nd) gives the probability difference, not "how many times more likely".
  • Wrong base probability: Using 5/2 instead of 5/(5 + 2) = 5/7. A ratio of 5 : 2 has 7 parts in total.
  • Unfinished ratio working: Writing just 5⁴ = 625 or 5² × 2² = 100 without showing the division 625 ÷ 100 scores 0 marks.

Topics

Probability · Ratio, proportion and rates of change · Number · 3.5 Probability · 3.3 Ratio, proportion and rates of change · 3.1.2 Fractions, decimals and percentages

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.