An event that consists of precisely one outcome. It's the most granular form of event you can observe in the chaotic dance of probability. Think of it as the universe exhaling a single, solitary breath. They're also known, rather starkly, as atomic events or, if you prefer the poetic, a sample point.

This concept is a cornerstone of probability theory, a field that attempts to quantify the unpredictable. It’s part of a larger series on statistics, delving into the mechanics of probability theory itself. We touch on the absolute bedrock: Probability, the fundamental Axioms that govern it, and the philosophical underpinnings of Determinism versus Indeterminism, where the very notion of a Deterministic system crumbles in the face of inherent Randomness.

Within this framework, we explore the building blocks: the Probability space, the all-encompassing Sample space, and the individual Event. We consider Collectively exhaustive events, where every possibility is covered, and the stark simplicity of the elementary event. Then there's Mutual exclusivity, where events cannot coexist, and the singular Outcome. We even look at the mathematical abstraction of a Singleton, a set containing just one element, mirroring the elementary event. The very act of observation, the Experiment, is dissected, from the simple Bernoulli trial to the complex Stochastic process. And, of course, the distributions that arise from these trials – the Bernoulli distribution, the ubiquitous Binomial distribution, the sprawling Exponential distribution, the familiar Normal distribution, the heavy-tailed Pareto distribution, and the ever-present Poisson distribution. We also consider the Probability measure itself, the Random variable that maps outcomes to numbers, and the Bernoulli process that describes sequences of such trials. The distinction between Continuous or discrete variables is crucial, as is understanding the Expected value and Variance. We trace paths with Markov chains and Random walks, observing the Observed value of these processes.

Beyond the singular, we examine relationships: Complementary events that complete the whole, Joint probability of multiple events, Marginal probability of individual events within a larger context, and the critical concept of Conditional probability. We ponder Independence and its more nuanced cousin, Conditional independence, and the elegant Law of total probability, the powerful Law of large numbers, and the transformative Bayes' theorem. Even Boole's inequality finds its place in this intricate web. Visual aids like Venn diagrams and Tree diagrams help us navigate these concepts.

The Singular Nature of Elementary Events

An elementary event, to put it bluntly, is an event that contains only one possible outcome. It’s the universe at its most distilled, the absolute smallest unit of possibility within a given sample space. In the cold, precise language of set theory, this translates to a singleton – a set containing a single element. For simplicity, and because the distinction often becomes blurred in practice, we frequently treat the elementary event and its corresponding outcome as interchangeable. It’s like saying "the red ball" when you mean "the event of picking the red ball."

Let's dissect this with some examples, because abstraction alone is rarely satisfying.

• Counting the Unseen: Imagine you're counting something, anything, and your sample space is the set of all natural numbers: $S = \{1, 2, 3, \ldots\}$. An elementary event here would be observing exactly the number 5, represented as the set $\{5\}$. Or perhaps observing the number $k$, where $k$ is some specific natural number. Each individual number is a solitary outcome, a single point in an infinite expanse.

• The Coin Toss Conundrum: Consider the classic coin toss, but let's make it twice. The sample space is $S = \{HH, HT, TH, TT\}$, where H signifies heads and T signifies tails. In this scenario, the elementary events are:

  • $\{HH\}$: Getting heads on both tosses.
  • $\{HT\}$: Getting heads on the first toss and tails on the second.
  • $\{TH\}$: Getting tails on the first toss and heads on the second.
  • $\{TT\}$: Getting tails on both tosses. Each of these is a singular sequence of results, a unique path through the possibilities.

• The Continuous Spectrum: Now, let's venture into the unsettling territory of continuous variables. Suppose you have a random variable $X$ that follows a normal distribution, with a sample space that spans all real numbers, $S = (-\infty, +\infty)$. Here, an elementary event would be observing exactly a specific real number, say $x$, represented as $\{x\}$. This is where things get… interesting. Because there are infinitely many real numbers, the probability of observing any single, specific real number is infinitesimally small, effectively zero. This is a crucial point: in continuous distributions, the probabilities assigned to individual elementary events are zero. They don't, by themselves, define the probability distribution. It's the intervals between these points that carry probability.

The Probability of a Solitary Occurrence

The probability assigned to an elementary event can range from zero to one, inclusive. It's a measure of its likelihood.

In a discrete probability distribution with a finite sample space, each elementary event is assigned a distinct, non-zero probability. This is where the probabilities add up neatly, like pieces of a puzzle.

However, as we saw with the continuous distribution example, individual elementary events in such distributions are assigned a probability of zero. They are points of no measure in a continuous landscape.

There are, of course, "mixed" distributions that defy such neat categorization. These distributions can possess both continuous stretches and discrete points. The discrete elementary events within these mixed distributions are often referred to as atoms or atomic events, and they can carry a non-zero probability. They are the exceptions that prove the rule, the solid ground in a shifting terrain.

It’s worth noting that under the more rigorous, measure-theoretic definition of a probability space, the probability of an elementary event isn't always explicitly defined. The set of events for which probability is assigned might be a specific σ-algebra on the sample space, rather than its entire power set. This is a level of detail usually reserved for advanced theoretical work, where the foundations are scrutinized with a fine-tooth comb.

Related Concepts

• Atom (measure theory): This is closely related to the concept of discrete elementary events in mixed distributions. An atom is essentially the smallest measurable set that possesses a positive measure.
• Pairwise independent events: While elementary events focus on a single outcome, pairwise independence deals with the relationships between pairs of random variables. It’s a step up in complexity, examining how the occurrence of one event influences another, but only in pairs.