# Using Probability Models

## Definition and Purpose

• Probability models provide simplified representations of real-world phenomena.
• They can help to derive predictions and make decisions based on uncertainty.
• The models offer ways to quantify and visualise the likelihood of various possible outcomes.

## How to Build Probability Models

• Identify the random variable – the variable outcome that is subject to uncertainty.
• Decide on the possible outcomes and establish their probabilities. Represent these in the model.
• Ensure that the sum of all possible outcomes equals 1, as this reflects absolute certainty that one of the stated outcomes will happen.

## Common Probability Models

• The Uniform distribution, where each possible outcome has an equal chance of occurring.
• The Normal distribution, characterised by its bell-shaped curve. The likelihood of outcomes decreases as you move away from the mean.
• The Binomial distribution, applied when there are multiple trials but each with only two possible outcomes.
• The Poisson distribution, used when the interest lies in the number of events that occur within a fixed interval of time or space.

## Applications in Real-life Contexts

• Probability models are used widely across various fields that handle uncertainty, such as Statistics, Finance, Engineering, and Ecology.
• In finance, risk models use probabilities to predict potential losses.
• Weather forecasts use probability to predict outcomes like rainfall or sunshine.

## Evaluating Probability Models

• Models should be tested against real-world data to assess their accuracy.
• There are formal statistical tests for this, like chi-squared tests and goodness of fit tests.
• Discrepancies between the model’s predictions and actual outcomes can highlight areas where the model needs to be refined.

## Limitations

• Probability models rely on assumptions which might not perfectly reflect reality.
• They may not account for exceptional events, which fall outside of normal probability parameters.
• A model is only as good as the quality and completeness of the data it is based upon. Misleading or incomplete data will affect the model’s accuracy and reliability.
• Over-reliance on models can be dangerous, they should be used as tools to support decision making, not to replace judgement.