Definition
Randomness
Randomness is uncertainty about which possible outcome of a random experiment will become the realised outcome. In a probability space , the model contains randomness when some event is neither impossible nor certain:
The probability measure quantifies this uncertainty without determining the realised outcome beforehand.
Randomness Depends on the Model
Randomness may arise because a physical process is intrinsically indeterminate or because the model omits information that would be needed for prediction. A coin toss, for example, may be governed by deterministic mechanics while remaining random in a model that does not track its exact initial conditions.
Probability theory treats both cases in the same mathematical way. It requires uncertainty within the model, not a claim that the underlying world is fundamentally indeterminate.
Randomness Still Has Regularity
Randomness concerns individual outcomes; it does not imply the absence of stable aggregate behaviour. Although each trial may be unpredictable, repeated comparable trials can exhibit frequencies that approach the probabilities specified by the model, as described by the law of large numbers.
Warning
Random does not mean equally likely. Equal probabilities are an additional modelling assumption, usually justified by symmetry or empirical evidence.