Statistical modellingEdexcel International A Level Maths: Revision notes

A mathematical model is a simplified description of a real situation that can be analysed using mathematics. In probability and statistics the model usually describes a situation using a random variable $X$ and a rule for its probabilities. For a fair six-sided die, $P(X=x)=\frac16$ for $x=1,\ldots,6$. A model cannot capture everything about reality. It works by making assumptions that simplify the situation.

Modelling follows a cycle:

1. Observe a real problem and decide what to find out.
2. Devise a model, stating assumptions and simplifying.
3. Use the model to make predictions (for example, expected frequencies).
4. Compare the predictions with real data or the real situation.
5. Refine the model if the comparison is poor, and repeat.
6. Report the findings, with the limits of the model.

Common modelling assumptions in probability are:

• outcomes are equally likely (a fair die, a fair coin, a random selection)
• trials are independent (one result does not affect the next)
• the probability is constant from trial to trial. For a spinner with five equal sectors, the model $P(X=x)=\frac15$ assumes that each sector is equally likely. For a bus that is late with probability $0.15$ on any day, the model assumes constant probability and independence between days.

If an experiment is carried out $n$ times and the model gives probability $p$ to an outcome, the expected frequency is $np$. The relative frequency is $\frac{\text{observed frequency}}{n}$. Example: a spinner modelled by $P(X=x)=\frac15$ is spun 200 times. The expected frequency of a 3 is $200\times\frac15=40$. If a 3 occurs 41 times, the relative frequency is $\frac{41}{200}=0.205$, close to $0.2$. With more trials, relative frequencies usually settle close to the model's probabilities, so large samples give better evidence.

To judge a model, compare its predictions with data. Small, patternless differences are consistent with random variation, so the model is reasonable. Large or systematic differences suggest it is unsuitable, and you should say which assumption may fail (for example, days not independent). Always comment in the context of the question. Refinements include changing probabilities, dropping an assumption such as independence, or collecting more data. Advantages of models: cheap, quick, allow predictions and 'what if' tests. Limitations: they simplify, so predictions can be wrong if the assumptions are unrealistic.