What is randomness?

What is randomness?

Leo Kee Chye

Monday, July 14, 2003 

An outcome chosen, or occurring without a discernible pattern, plan, or connection, according to a dictionary. Or is it? An innocuous conversation kindled my curiosity in the subject matter; subsequently, embarked me on an amateur quest where I ended up as much baffled as when I had started. The essay tells more.

When confronted with the concept of randomness, I effortlessly call to mind the images of tossing of a coin, rolling of a dice, and the turning of a roulette-wheel. Randomness means nothing more than the unpredictability of outcomes of any event, at least that was what I thought until I overheard this conversation. (Not in verbatim)

From then on, the conversation weighed heavily on my mind, like a splinter in my brain which I could not rid. What baffled me was how on earth a jackpot is able to churn out random outcomes. Contrary to conventional knowledge, modern jackpot does not operated on mechanical parts the likes of roulette wheel but on computational processes similar to those within a computer. But how could a deterministic machine churn out non-deterministic outcomes. It does not add up. Naturally, the splinter then extended itself to include randomness in other form of machineries : the pocket calculator that give a random number to 3 decimal places between 0 and 1 when you hit “RAN#”; the computer game that offers you hours of random fun.

After much scouring, I found enough literatures to confirm my suspicion that computers are essentially deterministic machines. Given the same set of inputs and instructions, the machines will surely generate the corresponding same outputs every time. However, with a complex computer algorithm (sets of computational instructions) and by feeding it with a seed number (an arbitrary number), the computer is therefore capable of generating a series of numbers that looks like random, behaves like random, but not random or commonly known as pseudo-random numbers. With the knowledge of the seed number and algorithm, it is possible to know all the output numbers beforehand.

My eyes brightened when I discovered this. Do you know what that could mean? Think of the all the jackpot machines lotteries, and sweepstakes in the world. There is a catch. Unless you find ways to steal a peep at the paper containing the seed number and algorithm, breaking the code could be close to impossible. That goes my get-rich scheme.

If those outcomes generated by deterministic machines are non-random, then can there be anything really random, in the true sense of the word? Outcomes from rolling the dice are often said to be random, provided the dice is not loaded. But are these outcomes considered unpredictable because it’s inherently unpredictable or it’s beyond the grasp of human’s present understanding of the event.

In March of 1993, Carolyn and Eugene Shoemaker and David Levy discovered a comet which they predicted would collide with Jupiter 16 months later. The collision did occur, thanks to their divination repertoire like mathematics and physics. In similar veins, the same laws of physics can be applied to augur the outcomes of the dice. But the sheer enormity of the possible physical causes and effects – the viscosity of air; the force of toss; direction of throw – assumed by the dice frustrates our computational abilities, not that they are totally unpredictable. Perhaps, only time stands in the way before humans’ intelligence will exorcise the spirit of randomness from dice rolling forever. But the nagging question remains: Is there any truly random event?

The closest we can possibly get to true randomness comes from natural processes like the decay of radioactive material. Yet, the debate remains whether the event is inherently unpredictable or merely due our present ignorance.

French mathematician Laplace, so confident in Man’s faculty of reason, asserted that “Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.” (Laplace).

But if everything is so predictable, or going to be predictable, then what’s so random about randomness? Nothing, for randomness to men like Laplace means an expression for our ignorance with regards to the actual causation. And probability, the mathematic language of randomness, is a merely an expedient which could be progressively eliminated through the advance of knowledge.

That position seems unassailable if one truly believe in a clock-like universe. Yet this stance has somewhat shaken with the advent of quantum mechanics which include an uncertainty principle that states it is impossible to determine both the position and momentum of a particle at the same time. The jury is still out.

Interestingly, the role of randomness, instead of diminishing, increases with our knowledge and its mathematical other-half, probability, has found its way into nearly every discipline.

Probability continues to bedevil us. Though pathetically hopeless in predicting outcomes in a single random event, it is capable of describing uncanny regularity for a large enough outcomes of the same event. For instance, we are quite inept in predicting “head” or “tail” for a toss of coin. Yet, we know that when we increase our tosses, the total outcome of “heads” should roughly equal to the total outcome of “tails”. From this simple observation, mathematicians have developed “law of large numbers”, “normal law, “central limit theorem” etc to give a face to randomness. The values of sample averages of a event, normal or non-normal, should take on the shape of a normal distribution curve, the ubiquitous bell-shape curve we learnt by rote in school, when the size of the sample average is large for non-normal population.

If randomness is a measure of our ignorance, it seems this ignorance not only can be measured but also display uncanny order, an order that emerged from chaos. However, these so-called laws do not necessitate the values of sample averages will distribute to the shape of a normal distribution or the sample averages will tend towards the mean when the size increases. Since we could not conduct experiment ad infinitum to verify our claim, we have to contend ourselves these laws are based on our limited observation, which may or may not be true. I do not see this as a major problem since physical laws are also developed based on limited observation.

But unlike physical laws which are confined to its proper domain, these statistical property of probability are applied in many branches of science, as well as economics, finance and other social sciences, whenever or wherever the subject matter displays randomness (or what we think is random, whatever that means). This requires making a leap of faith which most practitioners unblinkingly take for the inviolable law of nature.

In recent decades, “nonlinearity” theories, like Chaos and Neural networks, gain popularity in finance and economics forecasting. Nonlinearity means the outcomes are not proportionate to the cause. Therefore, in a dynamic system, the sample averages approaching the mean when the size increases is absurd since the mean is constantly in a state of flux. The bell-curve is merely an illusion.

Still, the question remains: what is really randomness? Though it looms large in everyday life, when we often use it without a second thought, randomness vanishes upon closer scrutiny.

It seems the probability of answering the question is close to zero, given my intellectual incompetence. And the probability of this endeavour eating into my time with no prospect of any monetary payoff in the future is approaching one. I guess I have to stop my quest here and meantime tolerate the splinter in my brain.

• Laplace, Pierre Simon De, (1812-1820), “Theorie Analytique de Probabilites: Introduction,” v. VII, Oeuvres