randomness

Randomness refers to the absence of a predictable pattern in events or sequences. In probability theory, a random process produces outcomes according to specified probabilities, rather than determined by a fixed rule. A sequence may be generated deterministically by an algorithm yet still appear random to an observer; such sequences are called pseudo-random. In formal discussions, randomness is studied from multiple perspectives, including statistical randomness, algorithmic randomness, and physical randomness.

Statistical randomness is assessed by how well observed frequencies match theoretical probabilities and by the absence of detectable patterns. Entropy measures uncertainty in a system, with higher entropy indicating greater randomness. Algorithmic randomness (Kolmogorov complexity) characterizes a sequence as random if there is no shorter computer program that produces it. No finite sequence can be proven random in the strictest sense, but sequences that resist compression are considered random in this framework. Practical tests for randomness include statistical test suites that check distributional properties and independence.

Randomness sources fall into true randomness and pseudo-randomness. True randomness originates from physical processes such as quantum measurement, radioactive decay, or thermal noise. Pseudo-random number generators use deterministic algorithms and a seed to produce long sequences that pass statistical tests; their unpredictability relies on computational assumptions. Cryptographic applications require strong unpredictability and resistance to reverse-engineering.

Applications span statistics, simulations (Monte Carlo methods), and decision making under uncertainty. In cryptography, randomness is essential for keys and nonces. Philosophically, randomness raises questions about free will and the nature of physical law, while practical use seeks a balance between reproducibility and unpredictability.