From Wikipedia on 11/4/12
Randomness means different things in various fields. Commonly, it means lack of pattern or predictability in events.
The Oxford English Dictionary defines “random” as “Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard.” This concept of randomness suggests a non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible pattern or combination.
Applied usage in science, mathematics and statistics recognizes a lack of predictability when referring to randomness, but admits regularities in the occurrences of events whose outcomes are not certain. For example, when throwing two dice and counting the total, we can say that a sum of 7 will randomly occur twice as often as 4. This view, where randomness simply refers to situations where the certainty of the outcome is at issue, applies to concepts of chance, probability, and information entropy. In these situations, randomness implies a measure of uncertainty, and notions of haphazardness are irrelevant.
Many scientific fields are concerned with randomness:
Algorithmic probability
Chaos theory
Cryptography
Game theory
Information theory
Pattern recognition
Probability theory
Quantum mechanics
Statistical mechanics
Statistics
Randomness versus unpredictability
Randomness, as opposed to unpredictability, is an objective property. Determinists believe it is an objective fact that randomness does not in fact exist. Also, what appears random to one observer may not appear random to another. Consider two observers of a sequence of bits, when only one of whom has the cryptographic key needed to turn the sequence of bits into a readable message. For that observer the message is not random, but it is unpredictable for the other.
One of the intriguing aspects of random processes is that it is hard to know whether a process is truly random. An observer may suspect that there is some “key” that unlocks the message. This is one of the foundations of superstition, but also a motivation for discovery in science and mathematics.
Under the cosmological hypothesis of determinism, there is no randomness in the universe, only unpredictability, since there is only one possible outcome to all events in the universe. A follower of the narrow frequency interpretation of probability could assert that no event can be said to have probability, since there is only one universal outcome. Under the rival Bayesian interpretation of probability, there is no objection to useing probabilities to represent a lack of complete knowledge of outcomes.
Some mathematically defined sequences, such as the decimals of pi mentioned above, exhibit some of the same characteristics as random sequences, but because they are generated by a describable mechanism, they are called pseudorandom. To an observer who does not know the mechanism, a pseudorandom sequence is unpredictable.
Chaotic systems are unpredictable in practice due to their extreme sensitivity to initial conditions. Whether or not they are unpredictable in terms of computability theory is a subject of current research. At least in some disciplines of computability theory, the notion of randomness is identified with computational unpredictability.
Individual events that are random may still be precisely described en masse, usually in terms of probability or expected value. For instance, quantum mechanics allows a very precise calculation of the half-lives of atoms even though the process of atomic decay is random. More simply, although a single toss of a fair coin cannot be predicted, its general behavior can be described by saying that if a large number of tosses are made, roughly half of them will show up heads. Ohm’s law and the kinetic theory of gases are non-random macroscopic phenomena that are assumed random at the microscopic level.