Mathematical models in probability and statistics
A model is a simplified description of a real situation, built so that probabilities and predictions can be calculated. It is judged by how well its predictions match what really happens.
Building and testing a model
- Observe the real situation and decide what matters.
- Make assumptions that simplify it.
- Use the model to work out predictions.
- Compare the predictions with real data.
- If they do not agree, change the assumptions and try again.
Common modelling assumptions
| Situation | Typical model | Assumption made |
|---|---|---|
| A fair die or coin | Each outcome equally likely | The die or coin is unbiased |
| A random selection | Every item has the same chance of being chosen | The selection really is random |
| Repeated trials | The result of one trial does not affect the next | Trials are independent |
| A measured quantity such as height | Roughly bell-shaped | The values are symmetrical about a mean |
| A linear relationship | A straight line fits the points | The change in one variable is steady as the other changes |
Commenting on a model
A common question gives a model and asks for a comment, or asks why it might not be suitable. A good answer has three parts.
- Name the assumption.
- Say whether it is likely to be true in this situation.
- Say what difference it makes to the result.
Independence is unlikely. If there is a long road closure or bad weather, the bus is more likely to be late on one day if it was late the day before. The probability of being late would be higher on those days than 0.2, so the model would underestimate the chance of a run of late days.
The model assumes sales stay at the weekly mean. It ignores differences between days, seasons and holidays. A single week is also a small sample. The prediction could be well away from the real total, so a longer record, split by season, would improve it.
Samples and populations
A population is all the items of interest and a sample is the part of it that is measured. A model built from a sample is only reliable if the sample represents the population.
Discrete and continuous
Data from counting (the number of goals) are discrete and take separate values. Data from measuring (a time or a mass) are continuous and can take any value in a range. The model and the diagram used depend on which it is.
These notes are AI generated and may contain mistakes. Check them against your syllabus.