Event (Probability)
Statistics & ProbabilityAn 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 that you are looking for in an experiment. It is what you are trying to calculate the probability of.
Example
When rolling a die, "rolling an even number" is an event. It includes outcomes $2$, $4$, and $6$. The probability of this event is $3/6 = 1/2$.
Key Insight
An event can include just one outcome (rolling a 3) or many outcomes (rolling an even number). Either way, we calculate the probability of the whole event.
Definition
An event is a subset of the sample space: a collection of one or more outcomes. Events can be simple (one outcome) or compound (multiple outcomes). The probability of an event is the sum of the probabilities of its individual outcomes.
Example
Rolling two dice: the event "sum equals $7$" includes outcomes $(1,6)$, $(2,5)$, $(3,4)$, $(4,3)$, $(5,2)$, $(6,1)$. $P(\text{sum}=7) = 6/36 = 1/6$.
Key Insight
Events can be combined using "and" (intersection) and "or" (union). The complement of an event is "not A." These set operations form the basis of probability calculations.
Definition
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. This restriction is necessary to avoid paradoxes (e.g., non-measurable sets like Vitali sets) that arise when trying to assign probabilities to arbitrary subsets.
Example
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
The concept of "probability zero event" is subtle: in continuous probability, many things can happen (have probability $0$ of occurring) yet together account for all outcomes. This is why "probability zero" does not mean "impossible."