Event

In probability theory, an event is a set of outcomes from the sample space, a subset whose collective probability we want to know, and it occurs whenever the experiment's actual outcome is one of the members of that set. When rolling a die, "the roll is even" is the event {2, 4, 6}, "the roll is greater than 4" is the event {5, 6}, and "the roll is a 3" is the event {3}. This usage is entirely different from the software meaning of an event as a notification or message; here an event is a pure mathematical object, a collection of outcomes, and the probability of the event is the total probability of the outcomes that belong to it.

The defining move is to treat an event as a subset, because that allows ordinary set operations to express logical statements about outcomes. The union of two events (A or B) occurs when at least one of them does, the intersection (A and B) when both do, and the complement (not A) when A fails. If a card is drawn from a standard 52-card deck, the event "heart" has 13 outcomes, the event "king" has 4, their intersection (the king of hearts) has 1, and their union has 13 + 4 - 1 = 16, which is the inclusion-exclusion correction that prevents double-counting the shared outcome. Venn diagrams are simply pictures of events as regions inside the rectangle of the sample space.

Events are classified as simple (a single outcome) or compound (several), and two special events anchor the whole system. The certain event is the entire sample space itself, with probability 1, and the impossible event is the empty set, with probability 0. Kolmogorov's 1933 axioms state exactly how probabilities of events must behave: they are never negative, the certain event has probability 1, and the probability of a union of mutually exclusive events is the sum of their probabilities. Every standard rule, such as P(not A) = 1 - P(A) and the addition rule above, follows from those three statements applied to subsets.

Not every subset is automatically an event once the sample space is infinite, which is why the formal theory restricts events to a sigma-algebra. A sigma-algebra is a collection of subsets that contains the whole space and is closed under complements and countable unions, so any logical combination of allowed events is still allowed. For finite or countable spaces one can safely treat every subset as an event, but for the real line, mathematicians such as Giuseppe Vitali showed in 1905 that pathological sets exist to which no consistent length can be assigned, so probability is defined only on well-behaved sets (the Borel sets and their completions). Practitioners rarely notice this restriction, but it is why probability theory can be rigorous for continuous quantities like heights or times.

A common pitfall is conflating an outcome with an event, and another is assuming that an event with probability zero is impossible. An outcome is a single element; an event is a set, even if the set has one member. In a continuous sample space, such as the exact time of a lightning strike, any specific value has probability zero yet some value does occur, so a zero-probability event is not logically impossible, and the one with probability 1 is not guaranteed (the difference is called almost-sure versus sure). Another mistake is to define events vaguely: "at least two" and "more than two" are different sets, and "at least one" is the complement of "none." Careful wording of the event is often most of the work in a probability problem.

Events are the working vocabulary of applied probability. In reliability engineering the event is "component fails before 1,000 hours" and system events combine them; in epidemiology it is "infected," "tests positive," or "hospitalized," with Bayes' rule linking the probabilities of events given one another; in A/B testing it is "user converts"; in machine learning, a classification error is the event that predicted and true labels differ, and the error rate is its probability. Stating the event precisely, as a subset of a clearly defined sample space, is what turns a vague question such as "how likely is trouble" into a computable number.

Read an event as a question about the experiment's outcome, phrased as the set of outcomes for which the answer is yes: probabilities attach to these sets, logic becomes set algebra, and nearly every probability mistake can be traced back to describing the set incorrectly.