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Note that the bias that "everything has a purpose or cause" is actually implicit in the expression "apparent lack of purpose or cause". Humans are always looking for patterns in their experience, and the most basic pattern seems to be cause/effect. This appears to be deeply embedded in the human brain, and perhaps in other animals as well. For example, dogs and cats often have been reported to have apparently made a cause and effect connection that strikes us as amusing or peculiar. For instance there is a report of a dog who, after a visit to a vet whose clinic had tile floors of a particular kind, refused thereafter to go near such a tiled floor, whether or not it was at a vet's.
To solve this 'problem', random events are sometimes said to be caused by chance. 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.
Traditionally, randomness takes on an operational meaning in natural science: something is apparently random if its cause cannot be determined or controlled. When an experiment is performed and all the control variables are fixed, the remaining variation is ascribed to uncontrolled (ie, 'random') influences. The assumption, again, is that if it were somehow possible to perfectly control all influences, the result of the experiment would be always the same. Therefore, for most of the history of science, randomness has been interpreted in one way or another as ignorance on the part of the observer.
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 nuclear decay[?] events in radioactive substances.
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.
For the purposes of simulation it is necessary to have a large supply of random numbers, or means to generate them on demand. Algorithmic information theory studies, among other topics, what constitutes a random sequence.
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 article for an example of the consequences of such a misestimate. 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 (qv); its key selection was also insufficiently random.
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Randomness is central to games of chance and vital to the gambling industry. The book The Eudaemonic Pie (ISBN 0595142362) is the story of an attempt to exploit non-randomness in the gaming industry. Some of the students involved have gone on to significant careers in physics and in computer science.
Random draws are often used to make a decision where no rational or fair basis exists for making a deterministic decision.
Deviations from randomness are often regarded by parapsychologists as evidence for the theories of parapsychology.