Randomness

randomchancerandomlyrandomizedrandom chancechaosnon-randomdisorderdisorderedrandom data
In the common parlance, randomness is the apparent lack of pattern or predictability in events.wikipedia
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Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
Individual random events are by definition unpredictable, but since they often follow a probability distribution, the frequency of different outcomes over numerous events (or "trials") is predictable. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events.

Random variable

random variablesrandom variationrandom
In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space.
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.

Stochastic process

stochastic processesstochasticrandom process
A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

Probability

probabilisticprobabilitieschance
In this view, randomness is a measure of uncertainty of an outcome, rather than its haphazardness, and applies to concepts of chance, probability, and information entropy.
When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes.

Applications of randomness

randomly
These and other constructs are extremely useful in probability theory and the various applications of randomness.
Randomness has many uses in science, art, statistics, cryptography, gaming, gambling, and other fields.

Monte Carlo method

Monte CarloMonte Carlo simulationMonte Carlo methods
Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, particularly in the field of computational science.
The underlying concept is to use randomness to solve problems that might be deterministic in principle.

Dice

dieDice rollingpolyhedral dice
For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4.
A die roll is made random by uncertainty in minor factors such as tiny movements in the thrower's hand; they are thus a crude form of hardware random number generator.

Random walk

random walkssimple random walkRandom walks on graphs
In the 1888 edition of his book The Logic of Chance, John Venn wrote a chapter on The conception of randomness that included his view of the randomness of the digits of pi, by using them to construct a random walk in two dimensions.
A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.

Randomized algorithm

probabilistic algorithmprobabilisticrandomized algorithms
In some cases, such randomized algorithms even outperform the best deterministic methods.
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic.

Chaos theory

chaoticchaoschaotic behavior
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.

Conspiracy theory

conspiracy theoriesconspiracy theoristconspiracy
Misunderstanding of this can lead to numerous conspiracy theories.
Some people prefer socio-political explanations over the insecurity of encountering random, unpredictable, or otherwise inexplicable events.

Determinism

deterministicdeterministcausal determinism
A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
The opposite of determinism is some kind of indeterminism (otherwise called nondeterminism) or randomness.

Quantum mechanics

quantum physicsquantum mechanicalquantum theory
According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random.
During a measurement, on the other hand, the change of the initial wave function into another, later wave function is not deterministic, it is unpredictable (i.e., random).

Cleromancy

casting lotscast lotslots
Cleromancy uses the casting of bones or dice to reveal what is seen as the will of the gods.
Cleromancy is a form of sortition, casting of lots, in which an outcome is determined by means that normally would be considered random, such as the rolling of dice, but are sometimes believed to reveal the will of God, or other supernatural entities.

Randomization

randomizedrandomizingrandomisation
Allotment is now restricted to selecting jurors in Anglo-Saxon legal systems, and in situations where "fairness" is approximated by randomization, such as selecting jurors and military draft lotteries.

Predictability

unpredictabilityunpredictablepredictable
In the common parlance, randomness is the apparent lack of pattern or predictability in events.

Random walk hypothesis

random walkrandomrandom walk model
The random walk hypothesis considers that asset prices in an organized market evolve at random, in the sense that the expected value of their change is zero but the actual value may turn out to be positive or negative.
The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.

Random sequence

Randomat randomrandom binary sequence
Random variables can appear in random sequences.

Entropy (information theory)

entropyinformation entropyShannon entropy
In this view, randomness is a measure of uncertainty of an outcome, rather than its haphazardness, and applies to concepts of chance, probability, and information entropy.

Random number generation

random number generatorrandom numberrandom numbers
Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, particularly in the field of computational science.
A random number generator (RNG) is a device that generates a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance.

Random number table

Results would sometimes be collected and distributed as random number tables.
Random number tables have been used in statistics for tasks such as selected random samples.

Pseudorandom number generator

pseudo-random number generatorPRNGpseudorandom
Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, particularly in the field of computational science.
The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's seed (which may include truly random values).

Statistical randomness

randomrandom numbersstatistically random
These methods may vary as to how unpredictable or statistically random they are, and how quickly they can generate random numbers.
Statistical randomness does not necessarily imply "true" randomness, i.e., objective unpredictability.

Modern synthesis (20th century)

modern synthesismodern evolutionary synthesisevolutionary synthesis
The modern evolutionary synthesis ascribes the observed diversity of life to random genetic mutations followed by natural selection.

Probability space

probability measuresGaussian measureoutcomes
Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a probability space illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%).
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.