Event (probability)
High school
Consider a random experiment, such as rolling two dice, whose outcomes are pairs of values, such as $(1, 3)$. An event is a set of outcomes, such as “the sum of the dice is $4$,” which is the set $A = \setdef{(1, 3), (2, 2), (3, 1)}$. The event occurs if the outcome of the experiment is in the set.
Undergraduate (measure-theoretic probability)
Consider a probability space $(\Omega, \mathcal{F}, P)$. An event $A$ is a subset of the sample space $\Omega$ such that $A$ is an element of the σ-algebra $\mathcal{F}$. For each event, the probability measure $P$ assigns the probability that $A$ occurs in each experiment.
Properties
- An outcome $\omega$ and the event $\setdef{\omega}$ are different objects: an outcome is a result coming from a particular experiment, and an event is a set of results which describes which outcomes satisfy a property of interest.
- When $\Omega$ is uncountable, such as $[0, 1]$, it is impossible for every subset of $\Omega$ to be an event, because no probability can be assigned consistently to every subset.