In ordinary language, the word random is used to express apparent lack of purpose or cause. This suggests that, whatever is causing something, its nature is not only unknown but the consequences of its operation are also unknown. In most technical senses, randomness has an additional positive meaning related to some of the statistical properties of the observed. Thus, the landing location of water droplets from a waterfall will be random in the ordinary sense—who knows just what forces have applied to this or that droplet causing it to fall where it does? But in a statistical sense, droplet landing spots are not distributed randomly at all, depending on the scale of observation. Thus, all droplets are confined to a relatively small area about the base of the fall, and within that area have a distinctly non-uniform distribution. On the other hand, the sound intensity at this instant of any chosen frequency of electrical circuit noise (eg, 'hiss') is technically random as well as conventionally random.
In some applications, both randomness (as tested statistically) and unpredictability are required, as for instance in most cryptography uses. In other applications, such as many modeling or simulation applications, unpredictability is not only unnecessary, but may cause problems as for instance whilst repeating modeling runs during model 'acceptance tests'.
Sensibly dealing with randomness is a seriously hard problem in modern science, mathematics, psychology and philosophy. Merely defining it adequately for the purposes of this or that discipline has been quite difficult. Distinguishing between apparent randomness and actual randomness has been no easier, and additionally assuring unpredictability, especially against a well motivated party (in cryptographic parlance, the "Adversary"), has been harder still.
It is because of this bias that the absence of a cause seems problematic. See causation.
To solve this 'problem', random events are sometimes said to be caused bychance. Rather than solving the problem of randomness, this opens the gaping hole of the ontological status of chance. It is hard to avoid circularity by defining chance in terms of randomness.
With the advent of quantum mechanics, however, it appears that the world might be irreducibly random. According to the standard interpretations of the theory, it is possible (and in fact very, very easy) to set up an experiment with total control of all relevant parameters, which will still have a perfectly random outcome. The resistance to this idea takes the form of hidden variable theories in which the outcome of the experiment is determined by certain unobservable characteristics (hence the name "hidden variables").
Many physical processes resulting from quantum-mechanical effects are, therefore, believed to be irreducibly random. The best-known example is the timing of radioactive decay events in radioactive substances.
Deviations from randomness are often regarded by parapsychologists as evidence for the theories of parapsychology.
The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling but soon in connection with situations of interest in physics.
Statistics is used to deduce the underlying probability distribution of a collection of empirical observations.
Access to a source of high-quality randomness is absolutely critical in many applications of cryptography. For example, even a subtly non-random key choice may result in a complete break into a communications channel that was believed to have been secure and was relied upon to be so. See the Enigma machine and one-time pad articles for examples of the consequences of such mis-estimates. Keys used for the Enigma were non-random in many cases which made it possible for Allied cryptanalysts to break into the traffic with substantial consequences for the Nazi war effort. A similar thing happened in the Pacific Theater of WWII with the Japanese 'Purple' machine; its key selection was also insufficiently random. The key material used in the theoretically unbreakable one-time pad must be random and unpredictable lest the encryption technique become trivially breakable. Even a slight predictability of the key material used removes the one-time pad from the unbreakable category. The world's first programmable digital electronic computer was developed to attack a mechanical (and subtly non-random) implementation of the one-time pad.
