Try to double or triple or even x10 the number of dots on your piece of paper and you get "patterns" forming. Repeat the experiment 1 million times and you might find other patterns appearing.
I love thinking about the concept of randomness, too. Not so much from a mathematical perspective though.
When you flip a coin, the air, the way you flip, the way the coin is manufactured, all affect the outcome in ways we can't really calculate. When you roll a 10-sided die or 6-sided one, the manufacturing of it, the way some sides might have more density than others. The way a "10" is grooved means that side is ever so lighter than the side that has a "1". For sure all these affect the outcome, and, therefore, have a say in how random the coinflip or dice throw is. I've seen people manipulate dice throws, coin flips, measuring exactly how much strength to flip the coin, ensuring every throw has the same number of flips.
The way a dealer shuffles. The way a casino card stack is cut, the way the roulette wheels are oiled. The way the metal ball hits when it's thrown on the wheel.
How random is random? It's a lovely question!
And it is also lovely how a seemingly simple gambling question quickly escalated into a metaphysical one.
If we look for absolute randomness, I am afraid there is none. Not in a toss of a coin or a roll of a dice, not even in the movements of the atoms in a single object, and not in the workings of the universe. Everything factored in, there is no result which we can consider an absolute random one.
Or perhaps, just like in freedom, justice, joy, and everything else, there are no absolutes. That applies to randomness as well.