Event (Probability)
An event is a specific outcome or set of outcomes from a probability experiment that we are interested in.
Definition
An event is a specific result or group of results you are looking for in an experiment, the thing you are trying to calculate the probability of. It is a subset of the sample space: a collection of one or more outcomes, either simple (one outcome) or compound (multiple outcomes), and the probability of an event is the sum of the probabilities of its individual outcomes. Formally, in measure-theoretic probability, an event is an element $A$ of the sigma-algebra $\mathcal{F}$ on the sample space $\Omega$; not every subset of $\Omega$ is necessarily an event, only measurable subsets are, a restriction necessary to avoid paradoxes (e.g., non-measurable sets like Vitali sets) that arise when trying to assign probabilities to arbitrary subsets.
Example
When rolling a die, "rolling an even number" is an event that includes outcomes $2$, $4$, and $6$, with probability $3/6 = 1/2$. Rolling two dice, the event "sum equals $7$" includes outcomes $(1,6)$, $(2,5)$, $(3,4)$, $(4,3)$, $(5,2)$, $(6,1)$, so $P(\text{sum}=7) = 6/36 = 1/6$. For a continuous uniform distribution on $[0,1]$, $P(\{x\}) = 0$ for any single point $x$, yet $P([0,1]) = 1$: individual points are events (with probability $0$), but they collectively make up the entire sample space.
Key Insight
An event can include just one outcome (rolling a 3) or many outcomes (rolling an even number), and events can be combined using "and" (intersection) and "or" (union), with the complement of an event being "not A," forming the basis of probability calculations. The concept of a "probability zero event" is subtle: in continuous probability, many things can happen (have probability $0$ of occurring) yet together account for all outcomes, which is why "probability zero" does not mean "impossible."