In probability theory, an event is fundamentally a subset of the sample space, which is the collection of all possible outcomes of an experiment. Events can range from simple, like a single outcome, to complex, encompassing multiple outcomes. For instance, in a standard deck of 52 playing cards, drawing a single card means the sample space has 52 elements. An event could be drawing "The 5 of Hearts" (a single outcome), "A King" (four outcomes), or "A Spade"
(thirteen outcomes). Events are typically represented as sets, such as {1, 2, 3}, and can be visually depicted using Venn diagrams.
Defining Events in Finite and Infinite Sample Spaces
When the sample space contains only a finite number of outcomes, defining all subsets of this space as events works effectively. In such cases, the probability P of an event A, where each outcome in the sample space Ω is equally likely, can be calculated using the formula P(A) = |A| / |Ω|. Here, |A| represents the number of outcomes in event A, and |Ω| is the total number of outcomes in the sample space. This formula can be readily applied to various example events, such as those involving playing cards.However, this approach encounters difficulties when the sample space is infinite. Many standard probability distributions, like the normal distribution, have sample spaces that are the set of real numbers or a subset thereof. Attempts to assign probabilities to all possible subsets of real numbers lead to problems with "badly behaved" sets, specifically those that are nonmeasurable. To address this, it becomes necessary to limit attention to a more restricted family of subsets.
The Significance of Sigma-Algebras
For the standard tools of probability theory, including joint and conditional probabilities, to function correctly, it is essential to use a σ-algebra. A σ-algebra is a family of sets that is closed under complementation and countable unions of its members. This mathematical structure ensures that probabilities can be consistently assigned and manipulated. The most natural choice for a σ-algebra is the Borel measurable set, which is derived from unions and intersections of intervals. However, in practical applications, the larger class of Lebesgue measurable sets often proves more useful.In the general measure-theoretic description of probability spaces, an event is formally defined as an element of a selected σ-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the σ-algebra is not considered an event and, consequently, does not have a probability assigned to it. Nevertheless, with a properly specified probability space, all events of practical interest are indeed elements of the σ-algebra, ensuring their measurability and the ability to assign them probabilities.
