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Elementary event

In probability theory, an elementary event, also called an atomic event or sample point, is the fundamental unit of a probability space, defined as a singleton set containing exactly one outcome from the sample space, which represents all possible results of a random experiment.^[1]^[2]^[3] These events serve as the building blocks for constructing more complex events, which are subsets of the sample space comprising multiple elementary outcomes, allowing for the assignment of probabilities to broader scenarios in accordance with Kolmogorov's axioms.^[4]^[5] In discrete probability models, such as coin flips or dice rolls, each elementary event typically has an equal probability if the outcomes are equally likely, often denoted as \frac{1}{n} where n is the number of sample points.^[6]^[7] The concept is essential for defining the sigma-algebra of events and ensuring that probabilities are well-defined, non-negative, and sum to 1 over the entire sample space.^[8]

Definition and Context

Core Definition

In probability theory, an elementary event is a singleton subset of the sample space \Omega, which represents the universal set of all possible outcomes in a random experiment, and it corresponds to an indivisible outcome that cannot be decomposed into simpler events.^[9] Formally, if \Omega is the sample space, then an elementary event is denoted as \{\omega\} for some \omega \in \Omega.^[10] These events act as the atoms of the event space, serving as the basic building blocks from which all other events are formed by taking unions of such singletons.^[10] The concept of the elementary event gained prominence through Andrey Kolmogorov's axiomatic formulation of probability in 1933, where he explicitly identified these as the fundamental elements e of the set E (the space of elementary events), distinguishing them from composite random events that are subsets of E.^[10]

Relation to Sample Space

In probability theory, the sample space, denoted as \Omega, represents the universal set encompassing all possible outcomes of a random experiment, while elementary events correspond to the individual singleton subsets \{\omega\} for each \omega \in \Omega.^[11] These elementary events serve as the atomic units, capturing the most basic, indivisible results of the experiment.^[12] The structure of events builds upon these elementary events through the formation of a sigma-algebra \mathcal{F} on \Omega, which includes the empty set, \Omega itself, and is closed under countable unions, intersections, and complements.^[13] In finite sample spaces, \mathcal{F} is often the full power set of \Omega, consisting of all possible subsets, each of which is a finite union of elementary events.^[11] For infinite sample spaces, particularly continuous ones, the sigma-algebra is typically generated by a basis such as open intervals (Borel sigma-algebra), which includes the elementary events as measurable sets, though they are assigned probability zero. Events are measurable sets in this sigma-algebra, not necessarily countable unions of singletons.^[14] Regardless of the cardinality of \Omega, elementary events retain their indivisible nature, forming the foundational layer from which all composite events are derived.^[15] This relational structure underscores the prerequisite role of elementary events in establishing the event algebra of a probability space, enabling the subsequent definition of measurable sets and probability measures.^[16]

Probability Assignment

Probability Measure on Elementary Events

In probability theory, the assignment of probabilities to elementary events forms the foundational layer of a probability space, where an elementary event is a singleton subset {ω} for some outcome ω in the sample space Ω. The probability measure P is defined such that it maps each elementary event to a value in [0,1], ensuring that probabilities are non-negative and collectively normalize to unity across the sample space. This measure extends to more complex events through the principles of additivity, establishing a consistent framework for probabilistic reasoning.^[17] The axiomatic foundation for this probability measure stems from Kolmogorov's three axioms, which apply directly to elementary events in discrete settings. Specifically, the first axiom requires that P({ω}) ≥ 0 for every ω ∈ Ω, guaranteeing non-negativity as a core property of valid probabilities. The second axiom enforces normalization by stipulating that the sum of probabilities over all elementary events equals 1, i.e., ∑_{ω ∈ Ω} P({ω}) = 1, which ensures the total probability is conserved. These properties collectively define a valid probability distribution over the elementary events, serving as the building blocks of the sample space.^[17] Furthermore, the third axiom of countable additivity extends the measure to unions of disjoint elementary events. For a countable collection of disjoint elementary events {ω_i}, the probability of their union is the sum of their individual probabilities:

P\left( \bigcup_i \{\omega_i\} \right) = \sum_i P(\{\omega_i\})

This additivity principle allows the probability measure on singletons to propagate to arbitrary events within the sigma-algebra generated by the elementary events, maintaining consistency in both finite and countably infinite discrete cases. Non-negativity and normalization remain invariant, preventing negative or super-unitary probabilities and upholding the integrity of the measure.^[17]

Uniform vs. Non-Uniform Cases

In the uniform case, each elementary event \omega \in \Omega in a finite sample space \Omega is assigned equal probability, such that P(\{\omega\}) = \frac{1}{|\Omega|}.^[18]^[19] This assumption holds for scenarios like fair coin tosses, where the sample space \Omega = \{\text{heads}, \text{tails}\} yields P(\{\text{heads}\}) = P(\{\text{tails}\}) = \frac{1}{2}, or standard dice rolls with \Omega = \{1, 2, \dots, 6\} and each face equally likely at \frac{1}{6}.^[18]^[20] In contrast, the non-uniform case allows probabilities to vary across elementary events, provided they are non-negative and sum to 1 over \Omega.^[19]^[18] These assignments are modeled using a probability mass function (PMF), which specifies P(\{\omega\}) for each \omega.^[19] For instance, a biased coin might have P(\{\text{heads}\}) = 0.7 and P(\{\text{tails}\}) = 0.3, reflecting unequal likelihoods due to physical imperfections.^[19] For any event E as a union of elementary events in the discrete uniform case, the probability simplifies to P(E) = \frac{|E|}{|\Omega|}, where |E| counts the favorable elementary outcomes.^[18]^[20] Uniform assignments simplify inference by reducing probability computations to mere counting of outcomes, avoiding the need to sum disparate values from a PMF.^[18] However, many real-world scenarios, such as biased experiments or weighted sampling, necessitate non-uniform models to accurately capture varying likelihoods among elementary events.^[19]^[18]

Examples and Illustrations

Discrete Sample Spaces

In discrete sample spaces, which are finite or countably infinite sets of possible outcomes, elementary events correspond to the individual singleton outcomes that form the basic building blocks of the probability model.^[21] These spaces allow for the explicit listing of all outcomes, making it straightforward to identify and work with elementary events as the indivisible units from which more complex events are constructed.^[22] A classic example is the toss of a fair coin, where the sample space \Omega = \{H, T\} consists of two outcomes: heads (H) or tails (T). Here, the elementary events are the singletons \{H\} and \{T\}, each representing an atomic outcome of the experiment.^[21] In a uniform probability assignment, the probability of each elementary event is P(\{H\}) = P(\{T\}) = \frac{1}{2}.^[22] Another illustrative case is the roll of a fair six-sided die, with sample space \Omega = \{1, 2, 3, 4, 5, 6\}. The elementary events are \{1\}, \{2\}, \dots, \{6\}, each denoting the occurrence of a specific face value.^[23] Under uniform probability, the measure assigned to each is P(\{i\}) = \frac{1}{6} for i = 1, 2, \dots, 6.^[23] The discrete nature of these sample spaces enables the direct enumeration of elementary events, which in turn facilitates the calculation of probabilities for compound events through simple summation of the probabilities of the constituent elementary events.^[24] This approach is particularly valuable in finite cases, as it provides a concrete method to verify that the total probability sums to 1 across all elementary events.^[22]

Continuous Sample Spaces

In continuous sample spaces, the sample space \Omega is uncountable, typically consisting of all real numbers within an interval or more complex sets, such as \Omega = [0,1] for a uniform distribution representing proportions or normalized times.^[25] Here, elementary events are singletons \{x\} for each x \in \Omega, but unlike discrete cases, these have measure zero under the probability measure.^[26] For the uniform distribution on [0,1], the probability P(\{x\}) = 0 for any specific x, as the total probability mass of 1 is distributed continuously across the interval, making the likelihood of exact points negligible.^[25] Probability in such spaces is assigned via probability density functions rather than directly to singletons; meaningful events are intervals or sets with positive length, where P([a,b]) = \int_a^b f(t) \, dt for density f(t) = 1 in the uniform case.^[27] Elementary events serve as idealized building blocks, but their zero probability reflects the infinite divisibility of the space, ensuring the axioms of probability are satisfied without assigning positive mass to uncountably many points.^[26] A similar structure applies to non-uniform continuous distributions, such as the standard normal distribution with density f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}, where \Omega = \mathbb{R} and each elementary event \{x\} again satisfies P(\{x\}) = 0, despite the density f(x) providing the rate of probability accumulation around x.^[28] Probabilities are computed over intervals, like P(a < X < b) = \int_a^b f(x) \, dx, highlighting how the density informs event likelihoods without contradicting the zero-probability singletons.^[29] This zero-probability nature poses a conceptual challenge: elementary events in continuous spaces are not directly observable or practically distinguishable, as real measurements involve intervals due to precision limits.^[26] Consequently, probability theory shifts emphasis to intervals or Borel sets as the basic measurable units, treating singletons theoretically while focusing empirical analysis on events with positive measure.^[27]

Distinctions from Other Concepts

Elementary vs. Compound Events

In probability theory, an elementary event consists of exactly one outcome from the sample space, making it the indivisible building block of all probabilistic analyses.^[3] These events, often denoted as singletons such as {1} in the context of rolling a die, serve as the atomic units upon which more complex structures are built.^[30] A compound event, by contrast, arises from the union of two or more elementary events, allowing it to encompass multiple outcomes. For instance, when rolling a fair six-sided die, the elementary events include {1}, {2}, {3}, {4}, {5}, and {6}, whereas the compound event of obtaining an even number is the union {2, 4, 6}.^[31] This construction enables the representation of broader scenarios, such as "success" in a binary trial or "heads or tails" in a coin flip, which cannot be captured by a single elementary event.^[32] The primary distinctions lie in their structure and properties: elementary events are inherently atomic and mutually exclusive, meaning distinct ones cannot occur simultaneously since they represent unique outcomes with no overlap.^[33] Compound events, however, are decomposable into their elementary components and may overlap with other events, permitting intersections or shared outcomes in more intricate probability spaces.^[34] This decomposability underscores a key conceptual role, where elementary events furnish the fine-grained resolution required to define and calculate probabilities for compound events through systematic aggregation.^[35] When a compound event is the union of disjoint elementary events, its probability equals the sum of the individual probabilities of those components, reflecting the additivity property of probability measures.^[36] This relation ensures that probabilities remain consistent and computable, as the total likelihood distributes additively across non-overlapping basic outcomes.^[37]

Role in Sigma-Algebras

In the Kolmogorov axiomatic framework of probability theory, elementary events play a central role in the construction of sigma-algebras by serving as the foundational measurable sets that generate the collection of all permissible events. For a finite sample space \Omega, the elementary events are the singletons \{\omega\} for each \omega \in \Omega, and the smallest sigma-algebra containing these singletons is the power set $2^\Omega, which includes every possible union of elementary events and thus encompasses all subsets of \Omega as measurable events.^[38] This generation ensures that every event can be expressed as a disjoint union of elementary events, providing a complete algebraic structure for probability assignments in discrete models.^[39] In continuous sample spaces, such as \Omega = \mathbb{R}, the standard Borel sigma-algebra \mathcal{B}(\mathbb{R}) is employed, which is generated by the open intervals and contains all singletons \{x\} as measurable sets, since each singleton arises as a countable intersection of open intervals centered at x.^[39] Although the Borel sigma-algebra is not directly generated by the singletons alone—instead relying on intervals for generation—the elementary events remain measurable and form the atomic level from which more complex Borel sets are built through countable operations, ensuring that point outcomes are always included in the event space.^[38] This measurability is crucial, as it allows probability measures like the Lebesgue measure to assign probabilities (typically zero) to elementary events while defining measures for intervals and their combinations.^[39] Beyond basic spaces, elementary events underpin the sigma-algebras in abstract probability constructions, such as product spaces where the product sigma-algebra is generated by cylinder sets that specify outcomes in finitely many coordinates, effectively projecting elementary events from component spaces to form the basis for infinite-dimensional models.^[39] In settings like Markov chains, which can be viewed as processes on product spaces, these cylinder sets derived from elementary events enable the definition of measurable path spaces and transition probabilities.^[38] Within the standard Kolmogorov framework, elementary events thus consistently serve as the indivisible, measurable units that generate and populate the sigma-algebra, distinguishing classical probability from non-standard models like quantum probability, where events may not correspond to classical singletons due to non-commutative structures.^[40]

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