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How is randomness defined?

  1. Sep 27, 2011 #1
    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". However, if we intend randomness as events with equal frequency probability this can't be. Think for example of the frequency of symbol sequences of a crypted text file. So I'm wondering if there exists a rigorous definition of randomness in mathematics and/or physics which can be interpreted as the above definition?
  2. jcsd
  3. Sep 27, 2011 #2
    The symbol frequencies are pseudorandom, not truly random.
  4. Sep 27, 2011 #3
    Yes, and the question could be rephrased as: "how does someone distinguish between pseudorandomness and randomness?"
  5. Sep 27, 2011 #4
    In CS, it is common to distinguish between probabilistic and non-deterministic processes.

    A probabilistic process usually can generate events which satisfy the law of large numbers. If you run it a large number of times, you'll observe the statistical distribution of the chances of the individual events occurring.

    A non-deterministic process can generate events at will, there is only choice and no probability involved.

    The difference is best explained with a probabilistic 50/50 coin flip and a non-deterministic coin flip. The first will satisfy the law of large numbers (LLN), and -with high probability- with ten flips you end up close to a five times heads up distribution. The latter doesn't abide the LLN and anything can happen with ten coin flips.

    The mathematical underlying models often used are probabilistic and non-deterministic automata.

    (Non-deterministic automata are interesting in CS since, in general, software isn't probabilistic but non-deterministic. Checking for bugs, for example, with probabilistic means makes sense but cannot prove the absence of them.)

    [ I wish I could explain it better but even on the Wikipedia pages non-determinism and probability are conflated terms, which is a no-go area in CS theory.]
    Last edited by a moderator: Sep 27, 2011
  6. Sep 27, 2011 #5
    What is a "non-deterministic coin flip"? As I understand it coin flips always satisfy the LLN. Apart from this, what are the non-deterministic processes in the physical world? Some examples? (possibly NON quantum mechanical examples)
  7. Sep 27, 2011 #6
    Probabilism and Non-determinism are the mathematical made exact notions of mechanisms involving probability or possibility, respectively.

    The best example I know of to make the distinction apparent is a human. Let's say you have a switch where you can either light a red or a green bulb and a human operating that switch. Would you assign probabilities to that, like 50/50 or something? No, a human has a choice in operating it, and, for instance, might just always choose red.

    I don't know if non-deterministic processes in the real world really exist. To be honest, it is something which cannot be observed, non-determinism is just a technical notion which comes in handy in CS.

    (To really observe non-determinism you would need to find a process where it just looks infeasible to assign probabilities to it. I.e., something which fluctuates that wildly that your best guess is that it is non-deterministic. It is noteworthy that QM, for instance, doesn't observe non-determinism, which makes it very likely that physicist are observing not even a probabilistic process, but an entirely causal/mechanical system. Uh, IMO.)
    Last edited by a moderator: Sep 27, 2011
  8. Sep 27, 2011 #7
    But a human can somehow simulate randomness and if I see only the light without knowing if it is a human or natural source modulating it, I won't be able to distinguish between deterministic or non-deterministic, a random or pseudo-random symbol sequence.

    Yes, precisely. That's why I'm wondering if there is any scientific method to conceive as randomness as: "Having no definite aim or purpose; not sent or guided in a particular direction". Despite what most believe there is none.
  9. Sep 27, 2011 #8
    Pseudo randomness sometimes can be discovered, like in those famous 2D plots of random generators. Apart from that, yah, you're correct.

    I don't follow what you're saying here? Randomness is usually associated with probabilism, therefor your definition, to me, reads more as a definition of non-determinism. And that cannot be observed, I think. (Except for finding a process where events occur in such a manner that there doesn't seem to be a feasible manner of assigning probabilities. That would be difficult; well, on the other hand, maybe life is exactly like that, and it isn't that hard.)
  10. Sep 27, 2011 #9
    Yes, the question was if it is possible to distinguish between the two determinisms. If the answer is no, then I think it is sufficiently clear now. Thanks.
  11. Sep 28, 2011 #10


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    A process is random if the available information is useless in predicting the next outcome.
  12. Sep 28, 2011 #11
    Well, this could be a good working definition in many cases but it makes randomness knowledge dependent. My capacity to make use of the information in predicting the evolution of a system depends from my understanding of it and the laws that rule it. If today I'm not able to do this and the process "appears" random, tomorrow I may have a better theoretical background and randomness "disappeares". So it would be a subjective category not an objective one as science requests.
  13. Sep 28, 2011 #12
    That's not true. Randomness does not mean the behavior of a system cannot be modeled. Stochastic processes can be modeled if we can assign probabilities. These processes are still random even though we have information which allow us to model them probabilistically.

    The next outcome (x) will have a probability p(x). The uncertainty associated with predicting that outcome can be assigned a measure: U=p(x)(1-p(x)). U will be maximal when p(x)=0.5 and approach 0 as p(x) approaches either 1 or 0. Often U is normalized: U= 4(p(x)(1-p(x)).
  14. Sep 28, 2011 #13

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    Just piling on, there's at least two problems here:
    - Useless is far too strong a term.
    - "Next outcome" rules out continuous random processes.

    A process is "random" if the future evolution of the process is not uniquely determined by any (knowable) set of initial data.

    To be picky, there is no reason to exclude processes whose evolution over time is uniquely determined by initial data. For example, given the probability space {4}, I guarantee that every time you randomly select from this space you will get a four.
  15. Sep 28, 2011 #14
    But the question was what is randomness, not what a stochastic process is. And all depends from what we mean by "useful". If U nears 0 it is because the event is very likely or very unlikely. The former case can't be taken as a definition for randomness, the latter perhaps only if p(x)=1/N with N->infinity the nr. of possible outcomes. But then it hardly can be said to be "useful" for predicting the next outcome.
  16. Sep 28, 2011 #15
    If you include the parenthesis then it makes again the notion of randomness knowledge-observer dependent, not an intrinsic behavior of phenomena. If you exclude it then I wonder what that process might be? The only process I can think of is in QM without hidden variables. But there we won't also find any definition of randomness.

    I think that all boils down to the conclusion that randomness isn't a universally defined scientific concept. Despite widespread belief, the concept of "randomness" is not a scientific but a subjective category, like "beauty" or alike, which only gives a sense of our ignorance not an intrinsic property of processes or things.
  17. Sep 28, 2011 #16
    I was responding specifically to the post I quoted which described randomness incorrectly. A stochastic process is a process that can be described by a random variable. Look up the definition of a random variable. As for the broader definition of randomness, look up the Kolmogorov definition. It's considered the most rigorous generally accepted definition as far as I know.
  18. Sep 28, 2011 #17

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    If you are after a formal definition, the wiki article on random variables is pretty good:

    You are going to have to understand measure theory before you can make sense of that definition. Knowing the axiom of choice won't hurt.

    Without that knowledge, descriptions of randomness are going to look like handwaving. Just because the lay description is a bit loosey-goosey doesn't mean that a formal definition doesn't exist.
  19. Sep 28, 2011 #18


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    that is correct. But in some physical phenomena there is no information set that improves predicatability. To call information dependence subjective I find wrong. It is not subjective but lawfully determined.
  20. Sep 28, 2011 #19
    Well, my understanding of a random variable/process is one in which

    individual values cannot be predicted, but these outcomes (their values) can

    only be described probabilistically, and this is supposed to be an intrinsic issue

    and not just knowledge-dependent.
  21. Sep 28, 2011 #20


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    Stochastic processes always involve knowledge since they have a history.
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