AQA A-Level Mathematics Paper 3, June 2025: Question 15

5 marks · Easy difficulty · Multi-step Problem

Calculate binomial probabilities for the number of vegan diners in a random sample of 25 and state an assumption required for the binomial distribution model.

Practise this question

Question

Question 15 states that the proportion of diners at a restaurant who are vegan is 20%, modelled by a binomial distribution. Part (a) asks: in a random sample of 25 diners, (i) find the probability that no diner is a vegan (1 mark); (ii) find the probability that fewer than 6 diners are vegan (1 mark); (iii) find the probability that at least 9 diners are vegan (2 marks). Part (b) asks to state one assumption necessary for the binomial distribution to be valid in the context of this question (1 mark).

Mark scheme

Show the mark scheme Mark scheme for Question 15: 15(a)(i) awards B1 for finding probability in range [0.00377, 0.0038], typical solution 0.00378. 15(a)(ii) awards B1 for probability in range [0.616, 0.62], typical solution 0.617. 15(a)(iii) awards M1 for P(X <= 8) = [0.953, 0.954] or P(X >= 9) = 1 - P(X <= 8), and A1 for AWFW [0.0467, 0.047]. 15(b) awards E1 for stating in context that the probability of a diner being vegan is constant, or diners being vegan is independent from one diner to the next, or there are only two types of diners (vegan or non-vegan). Must use words 'vegan' and 'diner'.

How to answer it

Binomial Probabilities and Modelling Assumptions

📋 What this question tests

This question assesses your ability to set up and evaluate probabilities from a binomial model using statistical calculator functions, as well as stating valid assumptions required for binomial modelling strictly in context.

  • Defining the random variable and its parameters: X ~ B(n, p) where n = 25 and p = 0.2 .
  • Evaluating point probabilities: P(X = 0) .
  • Translating inequalities correctly: "fewer than 6" means P(X ≤ 5) ; "at least 9" means P(X ≥ 9) = 1 - P(X ≤ 8) .
  • Stating modelling assumptions using the context of the question (referencing both diners and vegan status).
Part (a)(i) — 1 Mark

Find the probability that no diner is a vegan

Binomial Point Probability

📐 Calculation

  1. Define the model: Let X be the number of vegan diners, so X ~ B(25, 0.20) .
  2. "No diner is a vegan" corresponds to P(X = 0) .
  3. Using formula: (0.80)²⁵ = 0.00377789...
  4. Round to 3 significant figures: 0.00378 .

✅ Correct Answer

0.00378

AWFW (Anything Which Falls Within): [0.00377, 0.0038]

B1 for obtaining the correct probability in range.
Part (a)(ii) — 1 Mark

Find the probability that fewer than 6 diners are vegan

Binomial Cumulative Probability (Strict Inequality)

📐 Step-by-Step

  1. Interpret the wording: "Fewer than 6" excludes 6.
  2. Translate to discrete cumulative notation:
    P(X < 6) = P(X ≤ 5) .
  3. Using the cumulative distribution function (Binomial CD) on your calculator with x = 5, n = 25, p = 0.2 :
  4. P(X ≤ 5) = 0.61668...
  5. Round to 3 significant figures: 0.617 .

✅ Correct Answer

0.617

AWFW: [0.616, 0.62]

B1 for correct probability in range.

❌ Common Error

Entering x = 6 into your calculator gives P(X ≤ 6) = 0.780 , which scores 0 marks. "Fewer than 6" means strictly less than 6!

Part (a)(iii) — 2 Marks

Find the probability that at least 9 diners are vegan

Complement Rule with Binomial CD

📐 Step-by-Step

  1. Interpret the wording: "At least 9" includes 9, so find P(X ≥ 9) .
  2. Apply the complement rule for cumulative tables/calculator:
    P(X ≥ 9) = 1 - P(X ≤ 8)
  3. Find P(X ≤ 8) = 0.9532...
  4. Calculate difference:
    1 - 0.9532 = 0.0468 (or 0.04677... ).

✅ Correct Answer & Mark Breakdown

0.0468 (or 0.0467)

M1 (Method): For finding P(X ≤ 8) = [0.953, 0.954] , or writing 1 - P(X ≤ 8) . (Also condones finding P(X ≤ 9) = 0.983 ).

A1 (Accuracy): AWFW [0.0467, 0.047].

🧠 Exam Technique

Always write down intermediate values like 1 - P(X ≤ 8) = 1 - 0.9532 . If you make a slip on the final subtraction, you still secure the M1 mark.

Part (b) — 1 Mark

State one assumption necessary for the binomial distribution to be valid in context

Modelling in Context

✅ Any One of These Contextual Statements

  • Independence: Whether one diner is vegan is independent of whether any other diner is vegan.
  • Constant probability: The probability (or proportion) of a diner being vegan is constant (or always 20% / does not change).
  • Two outcomes: There are only two outcomes: a diner is either vegan or not vegan.
E1 (Explanation): Awarded for a valid assumption clearly stated in context.

❌ Common Errors & Mark Losses

  • Missing context: Writing simply "Events are independent" or "Probability is constant" scores 0 marks. You must refer to diners and vegan.
  • "Probabilities are independent": Stating that the probabilities are independent is mathematically incorrect; the events/diners are independent.
  • Fixed number of trials: Simply stating "there is a fixed number of diners" is ignored because n = 25 was explicitly set by the question sample size.

💡 The BINS Checklist for Binomial Models

  • Binary: Two possible outcomes (vegan or not vegan).
  • Independent: One diner's choice does not affect another's.
  • Number of trials fixed: Sample size is 25.
  • Success probability constant: p = 0.20 for every diner.

Topics

Statistics · N: Statistical distributions

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