This thought has been nagging me since I first saw the pic below (taken from here):



The image on the left features a uniform distribution (which many erroneously come to think of as random), while the points on the right image are allegedly distributed randomly (read, it is a random distribution). But if we think about it, we may come to the idea that random is not truly random at a higher level. Really, if you hit some random outcome, you could in fact expect more of such outcomes in close vicinity of that first outcome as the image above clearly reveals. So how truly random is what universally considered random?

I'm starting the thread in the Gambling discussion section because this domain (i.e. gambling) is where the idea of randomness and whether random is truly random have most applications and implications. Yeah, you thought it right, for gambling and our success at it