Randomness in the context of Discrete distribution

A disease cluster is an unusually large aggregation of a relatively uncommon disease (medical condition) or event within a particular geographical location or period. Recognition of a cluster depends on its size being greater than would be expected by chance. Identification of a suspected disease cluster may initially depend on anecdotal evidence. Epidemiologists and biostatisticians then assess whether the suspected cluster corresponds to an actual increase of disease in the area. Typically, when clusters are recognized, they are reported to public health departments in the local area. If clusters are of sufficient size and importance, they may be re-evaluated as outbreaks.

John Snow's pioneering investigation of the 1854 cholera outbreak in Soho, London, is seen as a classic example of the study of such a cluster.

Terrorism, in its broadest sense, is the use of violence against non-combatants to achieve political or ideological aims. The term is used in this regard primarily to refer to intentional violence during peacetime or in the context of war against non-combatants. There are various different definitions of terrorism, with no universal agreement about it. Different definitions of terrorism emphasize its randomness, its aim to instill fear, and its broader impact beyond its immediate victims.

Modern terrorism, evolving from earlier iterations, employs various tactics to pursue political goals, often leveraging fear as a strategic tool to influence decision makers. By targeting densely populated public areas such as transportation hubs, airports, shopping centers, tourist attractions, and nightlife venues, terrorists aim to instill widespread insecurity, prompting policy changes through psychological manipulation and undermining confidence in security measures.

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.

Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha, beta, and gamma decay. The weak force is the mechanism that is responsible for beta decay, while the other two are governed by the electromagnetic and nuclear forces.

Radioactive decay is a random process at the level of single atoms. According to quantum theory, it is impossible to predict when a particular atom will decay, regardless of how long the atom has existed. However, for a significant number of identical atoms, the overall decay rate can be expressed as a decay constant or as a half-life. The half-lives of radioactive atoms have a huge range: from nearly instantaneous to far longer than the age of the universe.

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources.

This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem).

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which

• the domain is the set of possible outcomes in a sample space (e.g. the set ${\displaystyle \{H,T\}}$ which are the possible upper sides of a flipped coin heads ${\displaystyle H}$ or tails ${\displaystyle T}$ as the result from tossing a coin); and
• the range is a measurable space (e.g. corresponding to the domain above, the range might be the set ${\displaystyle \{-1,1\}}$ if say heads ${\displaystyle H}$ mapped to -1 and ${\displaystyle T}$ mapped to 1). Typically, the range of a random variable is a subset of the real numbers.

Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup.

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause or prevent a tornado in Texas.

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, despite the deterministic nature of these systems, this does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:

In probability theory, an experiment or trial (see below) is the mathematical model of any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.

When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis.

Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence.

The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory.

In metaphysics, balance is a point between two opposite forces that is desirable over purely one state or the other, such as a balance between the metaphysical order and chaos — law by itself being overly controlling, chaos being overly unmanageable, balance being the point that minimizes the negatives of both.

More recently, the term "balance" has come to refer to a balance of power between multiple opposing forces. Lack of balance (of power) is generally considered to cause aggression by stronger forces towards weaker forces less capable of defending themselves. In the real world, unbalanced stronger forces tend to portray themselves as balanced, and use media controls to downplay this, as well as prevent weaker forces from coming together to achieve a new balance of power. In constructed worlds, such as in video gaming, where nearly all-powerful corporate interests strive to maintain a balance of power among players, players tend to be extremely vocal about what they see as unbalanced mechanics, providing the unbalance negatively affects them. Though the strong and unbalanced (or "overpowered") players commonly are vigorous in denial of any lack of balance, the comparative media equality among all player brings change quickly, to further a sense of balance.

Creativity techniques are methods that encourage creative actions, whether in the arts or sciences. They focus on a variety of aspects of creativity, including techniques for idea generation and divergent thinking, methods of re-framing problems, changes in the affective environment and so on. They can be used as part of problem solving, artistic expression, or therapy.

Some techniques require groups of two or more people while other techniques can be accomplished alone. These methods include word games, written exercises and different types of improvisation, or algorithms for approaching problems. Aleatory techniques exploiting randomness are also common.

A game of skill is a game where the outcome is determined mainly by mental or physical skill, rather than chance.

Alternatively, a game of chance is one where its outcome is strongly influenced by some randomizing device, such as dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from a container.

An abstract strategy game is a type of strategy game that has minimal or no narrative theme, an outcome determined only by player choice (with minimal or no randomness), and in which each player has perfect information about the game. For example, Go is a pure abstract strategy game since it fulfills all three criteria; chess and related games are nearly so but feature a recognizable theme of ancient warfare; and Stratego is borderline since it is deterministic, loosely based on 19th-century Napoleonic warfare, and features concealed information.

A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, numbered balls, or in the case of digital games random number generators. A game of chance may be played as gambling if players wager money or anything of monetary value.

Alternatively, a game of skill is one in which the outcome is determined mainly by mental or physical skill, rather than chance.

In probability theory, a probability space or a probability triple ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a dice.

A probability space consists of three elements: