# Modelling with Probability

## Modelling with Probability

# 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.