Subevents

Subevents are events within a probability space that are contained inside a larger event. Formally, if A is an event (a subset of the sample space), a subevent B is a subset of A. If B is not equal to A, B is called a proper subevent. Subevents are commonly used to refine or partition outcomes, allowing a more detailed description of what can occur within a broader scenario.

In probability, subevents obey standard measure properties. If B is a subevent of A (B ⊆ A), then P(B) ≤ P(A). If a collection of subevents {B1, B2, …} are disjoint and their union equals A, then the probability of A is the sum of their probabilities: P(A) = Σ P(Bi). Conditional probability has a straightforward form when B ⊆ A: P(B|A) = P(B) / P(A) provided P(A) > 0.

Examples help illustrate the concept. Suppose A is the event “the roll of a fair die is even.” The subevents B1 = {2}, B2 = {4}, and B3 = {6} are subevents of A, and P(Bi) = 1/6 for each. The union of B1, B2, and B3 equals A, and P(A) = 1/2.

Applications of subevents span modeling complex processes, organizing event trees, and performing stepwise probability calculations. They provide a way to analyze finer-grained outcomes within a larger event and facilitate reasoning about conditional relationships and partitions.