What randomness really is

Problem Detail: 

I'm a Computer Science student and am currently enrolled in System Simulation & Modelling course. It involves dealing with everyday systems around us and simulating them in different scenarios by generating random numbers in different distributional curves, like IID, Gaussian etc. for instance. I've been working on the boids project and a question just struck me that what exactly "random" really is? I mean, for instance, every random number that we generate, even in our programming languages like via the Math.random() method in Java, essentially is generated following an "algorithm".

How do we really know that a sequence of numbers that we produce is in fact, random and would it help us, to simulate a certain model as accurately as possible?

Asked By : user1892655

Answered By : Juho

The short answer is that no one knows what real randomness is, or if such a thing exists. If you want to quantify or measure the randomness of a discrete object, you would typically turn to Kolmogorov complexity. Before Kolmogorov complexity, we had no way of quantifying randomness of say a sequence of numbers without considering the process that spawned it.

Here's an intuitive example that was really bugging people back in the day. Consider a sequence of coin tosses. The outcome of one toss is either heads ($H$) or tails ($T$). Say we do two experiments, where we toss a coin 10 times. The first experiment $E_1$ gives us $H,H,H,H,H,H,H,H,H,H$. The second experiment $E_2$ gives us $T,T,H,T,H,T,T,H,T,H$. After seeing the outcome, you might be tempted to claim there was something wrong with the coin in $E_1$, or at least for some weird reason what you got is not random. But if you assume both $H$ and $T$ are as probable (the coin is fair), the probability of obtaining either $E_1$ or $E_2$ is equal to $(1/2)^{10}$. In fact, obtaining any specific sequence is as probable as any! Still, $E_2$ feels random, and $E_1$ does not.

In general, since Kolmogorov complexity is not computable, one can't compute how random say a sequence of numbers is, no matter what kind of claimed "totally random" process spawned it.

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Question Source : http://cs.stackexchange.com/questions/12136