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

The question

A factory produces light bulbs, and it is assumed that each bulb has a 5% chance of being defective. If a batch contains 100 bulbs, what is the probability that exactly 5 bulbs are defective? Critique the assumption of a constant defect rate. [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

• S1

  Identify the problem: We need to find the probability of exactly 5 defective bulbs in a batch of 100.

• S2

  Make assumptions: Assume each bulb has a 5% chance of being defective and that the defects are independent.

• S3

  Choose a probability distribution: Use the binomial distribution with n = 100 and p = 0.05.

• S4

  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 .

• S5

  Interpret results: Calculate the value using a calculator or software to get the final probability.

Model answer

Worked through, with each step tagged to the mark it earns.

1. S1

   Identify the problem: We need to find the probability of exactly 5 defective bulbs in a batch of 100.

2. S2

   Make assumptions: Assume each bulb has a 5% chance of being defective and that the defects are independent.

3. S3

   Choose a probability distribution: Use the binomial distribution with n = 100 and p = 0.05.

4. S4

   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 .

5. S5

   Interpret results: Calculate the value using a calculator or software to get the final probability.

6. Final answer: P(X = 5) ≈ 0.180

Common mistakes

• Assuming events are independent when they are not. — Always check if events can influence each other and adjust the model accordingly.
• Using a single probability for all items in a heterogeneous population. — Consider different subgroups and use appropriate probabilities for each subgroup.
• Assuming a constant defect rate over time when it changes. — Consider how probabilities might change and use more realistic assumptions.
• Overlooking the impact of dependent events on the model. — Identify and account for dependencies in the model.
• Using overly complex models without justification. — Balance simplicity and complexity by justifying the choice of model based on the problem's requirements.
• Failing to critique assumptions in the final interpretation. — Always critique assumptions and discuss their potential impact on the model's accuracy.
• Assuming a fair coin without checking for bias. — 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. — Choose an appropriate distribution that accounts for dependencies between events.

Where the marks go

• Full worked solution (all marking points)4 marks

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