# What is randomness?

In probability and statistics, the concept of "random" and "randomness" are frequently used. Often the concept of a random variable is used to model events that occur due to chance.

My question regards the term "random". What is random? Does randomness really exist?

I am curious what people that have a lot of experience in working with random events think and believe about randomness.

• Are you seeking an authoritative answer or a collection of different opinions? Although I don't think there's any question that this subject is on topic, a question has been raised concerning whether this thread should be made CW (Community Wiki), especially because few of the existing replies appear authoritative. – whuber Apr 11 '12 at 18:05
• Yes, I believe that this thread should be a CW as I'm seeking a collection of opinions. – Andrew Apr 11 '12 at 18:10
• Much like causality, it is what you define it to be. See a possible definition here: en.wikipedia.org/wiki/Algorithmically_random_sequence – JohnRos Apr 11 '12 at 18:20

Here's a deflationary theory: Something is random when its behaviour is modeled formally using the machinery of probability theory, an axiomatized bit of pure mathematics. So in a sense the answer to the first question is rather trivial.

In approaching the rather less well-posed question 'does randomness really exist?' it's helpful to ask yourself whether vectors 'really' exist. And when you have a view about that, asking yourself a) whether it's surprising or not that polynomials are vectors, b) whether and how we could be wrong about that, and finally c) whether, e.g. forces in physics are the things that vectors 'are' in the sense of the question. Probably none of these questions will help much understanding what's going on in the forum, but they will bring out the relevant issues. You might start here and then follow up the other Stanford Encyclopaedia entries on philosophy of probability and statistics.

There is a lot of discussion there, thankfully not much found around here, about the existence and relevance of 'actual' physical randomness, usually of the quantum variety some of which is (usefully) gestured toward by @dmckee in the comments above. There's also the idea that randomness as some sort of uncertainty. Within the minimal framework of Cox it can be reasonable to think of (suitably tidied up) uncertainties as being isomorphic with probabilities, so such uncertainties are, by virtue of that connection, treatable as if they are random. Clearly the theory of repeated sampling also makes use of probability theory, by virtue of which its quantities are random. One or other of these frameworks will cover all the relevant aspects of randomness that I've ever seen in these forums.

There are legitimate disagreements about what should and should not be modeled as random, which you can find under the banners Bayesian and Frequentist, but these positions only suggest but do not full determine the meaning of the randomness involved, just the scope.

• +1 for introducing many thoughtful concepts into the discussion. I would like to suggest it may help to maintain a sharper distinction between randomness and uncertainty: one leads to the other, but not vice versa, yet many people (obviously not you!) exhibit some confusion about the difference. We know that not all uncertainty comes from randomness, nor is all that is arbitrary or variable necessarily "random" in the technical sense employed in statistical practice. – whuber Apr 11 '12 at 15:22
• I guess you're identifying random with sampling variability, which is obviously fine. I was trying to separate three things: the probability theory, the things that vary in repeated sampling, and uncertainty about stuff. (A strong and controversial connection claimed for the connections between them that might interest you is Lewis's 'Principal Principle' from 'A Subjectivist’s Guide to Objective Chance'.) – conjugateprior Apr 11 '12 at 23:45
• Please don't read that much into my comment: I had no intention of identifying randomness with sampling variability! I just wanted to call (positive) attention to some of the points you make. To agree or disagree with them would require a lengthy detailed analysis. (To get a sense of the kind of analysis involved, the article at plato.stanford.edu/entries/chance-randomness/#4 is of interest. But please don't assume that I hold with all the assertions in that article just because I am drawing attention to it!) – whuber Apr 12 '12 at 13:32

If we assume that we are living in a deterministic (everything that happens is predetermined and given the same exact situation, the same exact things will happen), then there is no "random" at all.

In this case, "randomness" is merely used to represent what might happen given our limited knowledge. If we had perfect knowledge of a system, nothing would be random.

• "If we had perfect knowledge of a system, nothing would be random."... Very philosophical... So, the concept of randomness is only a useful approximation to the unobservable components of a system? – Macro Apr 11 '12 at 12:07
• Quantum mechanics is very clear on this (now that repeated test of Bell's inequality have been done): the world either really does have randomness in it or is constructed in such a way that you really can not have sufficiently complete knowledge to predict everything going forward. – dmckee --- ex-moderator kitten Apr 11 '12 at 14:26
• (Deterministic) Newtonian mechanics is also clear on this: random phenomena arise even in classical physical systems. Invoking determinism is interesting, and helps us understand better what ought to count as "random," but ultimately is tangential to discussions of randomness in statistical practice or theory. – whuber Apr 11 '12 at 15:15
• Well put @dmckee. I'll point out that, while most people believe Quantum Mechanics states without doubt that the world is non-deterministic, this is not actually true - that is just one interpretation of quantum mechanics (which happens to be the most popular), but there are other, deterministic interpretations out there. – BlueRaja - Danny Pflughoeft Apr 11 '12 at 16:21
• @BlueRaja-DannyPflughoeft: Pay attention to what I wrote: either there is non-determinism or there is non-local information and you can not have complete knowledge. There is no point in bringing the interpretation of quantum mechanics into the discussion because the situation is independent of which interpretation you choose. – dmckee --- ex-moderator kitten Apr 11 '12 at 16:26

My definition of random would be unpredictable, i.e. you can never know with 100% certainty the outcome of an event, although you might be able to put a bound of the range of possibilities. A simple example would be rolling a fair dice: you can never know exactly which number will come up with each roll, but you do know it will be one of the numbers 1 through 6.

• "Unpredictable" makes intuitive sense, but doesn't it need some refinement? If I am ignorant of the machinery of the heavens, then the phases of Venus will be unpredictable to me. Does that make the workings of the solar system "random"? (You could make a case either way, and in so doing, you would clarify what you really mean by "unpredictable.") – whuber Apr 11 '12 at 15:27
• This would imply that randomness is "subjective". Since one's predictability of the future varies with knowledge and tools. This would be closer to the Bayesian view point. – Memming Apr 11 '12 at 16:14
• If one isn't ignorant of the machinery, if in fact one has 100% knowledge of how the machinery works but this still isn't sufficient to accurately predict outcomes, then this gap or inability to forecast is unpredictability or randomness. Just as Popper said that nothing is actually true but only accepted as true until falsified, babelproofreader says randomness is true, absolute unpredictability and no model, even an 100% infallibly accurate one, is actually good enough to predict randomness. This gap between reality and perfect knowledge of the "system" behind it is randomness. – babelproofreader Apr 11 '12 at 21:48

I tend to prefer a probabilistic interpretation of randomness. An event is random if gaining any additional information does not help you predict its outcome. That is, the event is unconditionally random. Notationally:

$p(A|B) = p(A) \forall B$

To put it in concrete terms; if you believe that a die roll (A) is truely random, then knowing the exact physical state of the die as it is thrown (B) confers no additional predictive power on the outcome of the toss.

• This is an intriguing approach, but doesn't it get things reversed? Once we are certain about an event, no additional amount of information helps us predict it any better. When an event is random--say, whether $Y\gt 0$ for a bivariate normal variable $(X,Y)$--then additional information, such as the value of $X$ in this case, usually does "confer additional predictive power" by allowing us to replace $\Pr(Y\gt 0)$ by $\Pr(Y\gt 0|X)$. – whuber Apr 11 '12 at 16:28
• No, the notation is a shorthand where $p(Y)$ should be expanded as $p(Y=y)$. After the event has occurred, you know it with certainty, i.e. $p(Y|Y=y, B)$ is 1 for $Y=y$ and 0 otherwise. And, yes, knowing $B$ (or $X$) is usually predictive, but then $A$ wouldn't be truly random. – Lucas Apr 11 '12 at 16:37
• Therefore, randomness is only in the future. Once the event has occurred, we know its value and it is no longer random... even if it were random before. – Andrew Apr 11 '12 at 17:26
• @Andrew: This is probably pedagogical, but it's the process of generating the event that is random, not the event itself. The event is just a thing. – Lucas Apr 11 '12 at 17:38
• A section in the Wikipedia article on randomness might help clarify how predictability and randomness differ. – whuber Apr 11 '12 at 18:08