AQA A-Level Mathematics: Modelling with Probability and Critiquing Assumptions — mark scheme explained

The question

Mark scheme, decoded

How the examiner actually awards the marks on this topic.

Gradora's own decode of the marking approach — not the exam board's published mark scheme.

Model answer

A full-mark response to the question above, worked through step by step.

⏱Timing: For a 4-mark question, allocate approximately 3-4 minutes to ensure you have enough time to carefully consider each step and critique your assumptions.

1. Identify the problem: We need to find the probability of exactly 5 defective bulbs in a batch of 100.0 marks
2. Make assumptions: Assume each bulb has a 5% chance of being defective and that the defects are independent.1 mark
3. Choose a probability distribution: Use the binomial distribution with n = 100 and p = 0.05.1 mark
4. Calculate probabilities: The probability of exactly k successes in n trials is given by P(X = k) = C(n, k) × p k × (1 - p) n-k . For k = 5, this becomes P(X = 5) = C(100, 5) × 0.05 5 × 0.95 95 .1 mark
5. Interpret results: Calculate the value using a calculator or software to get the final probability.1 mark

Final answer: P(X = 5) ≈ 0.180

✓Work through every step correctly and you earn all 4 marks.

Common mistakes

• Assuming events are independent when they are not.

  Why it happens: Students often assume independence without considering the context, leading to incorrect probabilities.

  Fix: Always check if events can influence each other and adjust the model accordingly.

• Using a single probability for all items in a heterogeneous population.

  Why it happens: Students might overlook differences within the population, leading to biased results.

  Fix: Consider different subgroups and use appropriate probabilities for each subgroup.

• Assuming a constant defect rate over time when it changes.

  Why it happens: Students might not account for factors that affect the probability over time, leading to inaccurate predictions.

  Fix: Consider how probabilities might change and use more realistic assumptions.

• Overlooking the impact of dependent events on the model.

  Why it happens: Students might not recognize that dependent events can significantly affect the results, leading to incorrect conclusions.

  Fix: Identify and account for dependencies in the model.

• Using overly complex models without justification.

  Why it happens: Students might create complex models that are difficult to manage and interpret, leading to confusion.

  Fix: Balance simplicity and complexity by justifying the choice of model based on the problem's requirements.

• Failing to critique assumptions in the final interpretation.

  Why it happens: Students might not evaluate the limitations of their assumptions, leading to overconfidence in the results.

  Fix: Always critique assumptions and discuss their potential impact on the model's accuracy.

• Assuming a fair coin without checking for bias.

  Why it happens: Students might assume fairness without considering external factors that could affect the coin's behavior.

  Fix: Check for any potential biases or influences on the coin and adjust the model if necessary.

• Using a binomial distribution when events are not independent.

  Why it happens: Students might apply the binomial distribution without verifying the independence of events, leading to incorrect probabilities.

  Fix: Choose an appropriate distribution that accounts for dependencies between events.

Where the marks go

The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.

Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.

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