AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 17

3 marks · Medium difficulty · Multi-step Problem

Calculate the combined relative frequency of Heads from two sets of coin throws, and identify which sample gives the best estimate of probability with a reason.

Practise this question

Question

Question 17 provides a table showing Sid threw a biased coin 80 times with a relative frequency of Heads of 0.75, while Zak threw it 120 times with 72 Heads. Part (a) asks to work out the relative frequency of Heads for all 200 throws for 2 marks. Part (b) asks to tick one box among 'Sid's 80 throws', 'Zak's 120 throws', or 'All 200 throws' to indicate which results give the best estimate of the probability of Heads, and give a reason, for 1 mark.

Mark scheme

Show the mark scheme Mark scheme for Question 17: Part (a) gives M1 for 0.75 × 80 or 60 (oe), and A1 for 0.66, 132/200, 66/100, or 33/50 (oe fraction, decimal, or percentage). Part (b) awards B1 for ticking 'All 200 throws' along with a valid reason such as having the most throws, a bigger number of trials, or more data.

How to answer it

Relative Frequency and Estimating Probabilities

📋 What this question tests

This question assesses your ability to:

  • Calculate an absolute frequency (count) from a given relative frequency and sample size.
  • Combine data from two independent sets of trials to find an overall combined relative frequency.
  • Understand that a greater number of trials yields a more reliable estimate of probability.

Question 17 (a)

Combined Relative Frequency (2 Marks)

📐 Step-by-Step Calculation

  1. Find Sid's number of heads:
    Number of Heads = 0.75 × 80 = 60
  2. Find the total number of heads:
    Sid's heads + Zak's heads = 60 + 72 = 132
  3. Calculate overall relative frequency:
    Total heads ÷ Total throws = 132 / 200
  4. Convert to decimal or simplest form:
    132 / 200 = 66 / 100 = 0.66 (or 33 / 50 )

✅ Correct Answer & Mark Scheme

Acceptable answers:

  • 0.66
  • 132 / 200 (unsimplified is fully accepted!)
  • 66 / 100 or 33 / 50
  • 66%
Mark Breakdown:
• [M1] For calculating 0.75 × 80 or stating 60 (or implied by reaching 132).
• [A1] For 0.66 or equivalent fraction/percentage.

💡 Key Knowledge

  • Relative Frequency formula:
    Relative Frequency = Number of successful outcomes ÷ Total trials
  • Rearranging to find count:
    Number of successful outcomes = Relative Frequency × Total trials
  • Do NOT average relative frequencies: Because the sample sizes are different (80 vs 120), you cannot simply add the two relative frequencies and divide by 2!

❌ Common Errors to Avoid

  • Averaging the probabilities: Doing (0.75 + 0.60) ÷ 2 = 0.675 scores 0 marks because the weights (80 and 120) are unequal.
  • Misinterpreting Zak's data: Writing 80 / 120 = 0.66 scores 0 marks. Zak threw it 120 times and got 72 heads, not 80.
  • Overcomplicating simplification: The mark scheme says "Ignore simplification or conversion attempts after correct answer". Leaving it as 132 / 200 guarantees the mark without risking an arithmetic slip!

Question 17 (b)

Evaluating the Best Estimate (1 Mark)

✅ Correct Answer

Tick the box: ☑ All 200 throws

Valid Reasons (any one of):

  • "It has the largest number of throws / trials."
  • "200 is the most throws."
  • "More trials give a more reliable / accurate estimate."
  • "It includes all the data collected."
Mark Breakdown:
• [B1] For ticking 'All 200 throws' AND providing a valid reason mentioning the larger number of trials/data.

🧠 Exam Technique: How to Give a Valid Reason

  • Always link your reason to the sample size (number of trials).
  • Use comparative words like "most", "largest", or "greater number of trials".
  • Writing just one word like "Accurate" is not enough — explain why it is more accurate (because there are more trials).

❌ Examiner Pitfalls for Part (b)

  • Vague statements: Saying only "More accurate" without mentioning the number of trials gives 0 marks.
  • Probability misconceptions: Writing "More possibility it will land on heads" is incorrect because the coin itself has a fixed underlying bias regardless of how many times it is flipped.
  • Just restating the question: Writing "Amount of heads out of 200" does not explain why 200 is better than 80 or 120.

Topics

Probability · 3.5 Probability

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.