# Definition:Probability

## Informal Definition

The **probability** of an event is the likelihood that the event will occur (within a certain formally or informally understood context).

As such, a probability can range from $0$ (will never happen) to $1$ (is certain to happen).

Commonly, a probability is often stated in terms of a percentage.

## Formal Definition

Let $\mathcal E$ be an experiment.

Let $A \in \Sigma$ be an event in the event space $\Sigma$ of $\mathcal E$.

Let $\Pr$ be the probability measure of $\mathcal E$.

Then the **probability** of $A$ is the value $\map \Pr A$.

## Also denoted as

Some sources give the **probability** of $A$ as $\map P A$

## Comment

Amongst the non-mathematically-literate of the world's population, a common form of rhetorical emphasis (usually used when asserting a falsehood of a personal and delicate nature) is to state a greater-than-$100 \%$ probability of an event.

A typical example of this could be seen on the Maury Povich Show, as follows:

- Euniquea: "I am five hundred thousand percent certain that Twyvone is the father of mah bay-bee ..."
- Maury Povich: "Twyvone ... you are
*naaht*the father."

## Sources

- 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*: $\S 1.3$