Unlike chaotic systems, there is no deterministic rule to generate randomness. Such a process derives from an inherently probabilistic rule.. True randomness turns out to be very hard to define and to distinguish from pseudorandomness.
The unexpected seems to contain more information than the expected. Learning that the next term in the series 2,4,6,...was 8 gives you no information once you know the rule. Learning the the next term in the series 11, 37, 191, ...was 48 would give you much more information. But the extreme case of this unexpected information is a set of random numbers, where it is impossible to build up expectations.
Algorithmically speaking, for example, complexity is generally measured by compressibility, eg. by the shortest program that will cause a Turing machine to print out a number. randomness is considered to be incompressible, that is to say the description of the object is no more compact than the object itself.
K-flows, named after the Russian mathematician Andrei Kolmogorov are a measure of chaos. K- entropy is a measure of the average rate at which trajectories starting from points extremely close together are moving apart. In a chaotic system an initial deviation will soon become as large as the true "signal" itself. Calculators or computers that round off numbers, to no matter how many digits, rapidly lead to errors in equations that both expand the number of digits and are sensitive to infinitesimally small differences in the numbers. However, in these cases K is positive but not infinite. It is infinite in totally random paths.
In Aristotle's sense of the term, automaton means sheer random happening, and tyche refers to some cause and effect sequence outside the usual pattern of development. See also clinamen for a classical account of random events.
Darwinian evolution is based on the theory of random mutation combined with natural selection.