M: Probability: Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.

The idea

Modelling with probability means using probability models to represent real-world situations. You must critique the assumptions made in a model and discuss how more realistic assumptions would affect the model's predictions. For example, assuming a coin is fair gives a probability of 0.5 for heads, but if the coin is biased, the probability changes. When critiquing, identify each assumption, explain why it may not hold, and state whether the true probability would be higher or lower. For instance, assuming independent trials in a medical test may be unrealistic if the disease spreads, leading to clustered results. More realistic assumptions often make the model more complex but more accurate. You should be able to suggest improvements and describe the likely direction of change in the calculated probabilities.

What earns the marks

• Identify the assumptions made in a given probability model, such as independence, equal likelihood, or constant probability.
• Critique each assumption by explaining why it might not be valid in the real-world context.
• Describe the likely effect of relaxing an assumption on the calculated probabilities, stating whether they would increase or decrease.
• Suggest more realistic assumptions and explain how they would alter the model or its predictions.
• Evaluate the trade-off between simplicity and accuracy when choosing a probability model.
• Communicate critiques clearly, using appropriate probabilistic language and referring to the context.

Common mistakes and how to put them right

• Failing to identify any assumptions and instead just describing the model.

  Put it right: Explicitly list assumptions such as independence or fairness before critiquing.

• Critiquing an assumption without stating the effect on the probability.

  Put it right: Say whether the probability would be higher or lower and why.

• Assuming that more realistic models are always better without considering practicality.

  Put it right: Discuss both accuracy and simplicity, and note that added complexity may not be justified.

• Using vague language such as 'it might change' without direction.

  Put it right: Be specific, e.g. 'the probability would increase because...'.

Exam tips

• Read the context carefully and list all assumptions before writing your critique; this ensures you address the modelling aspect fully.
• Use phrases like 'assuming independence' or 'assuming a fair coin' to make your assumptions explicit.
• When suggesting a more realistic assumption, always state the likely effect on the probability, using comparative language such as 'higher' or 'lower'.
• Link your critique back to the original problem to show you understand the practical implications.