Is There Really Any Order?

So, bear with me here. While I was in college, there were these "litmus tests" that were in some of our books to see if a person's "critical thinking" skills were good enough to operate on the level of some of the material. One of these tests was based on the understanding of one of Plato's dialogues, "The Euthyphro". In it, the question of piety was raised, i.e., is something pious because God loves it, or does God love something because there's already some inherent quality within the object that makes it "pious". This got me thinking about these exercises, and I've made a few of them in the same vein that I thought were interesting. Here's one.

If something is random, that means that it's able to become anything. When we think of randomness, it often evokes the idea of out-of-order, or mixed up; there's usually no symmetry in the things we consider random. By definition, however, if something is completely random, that assumes that it could also be symmetrical. For example, if ten numbers were placed in a hat, and each was drawn once for an infinite number of times, eventually the sequence of 1-10 in numerical order would be drawn, even though the numbers are taken at random. Following from this, it seems that any amount of order could come out of randomness. Take the image of a puzzle; if 50 puzzle pieces were thrown on the floor in a pile, very few, most probably none of them, would land close to their mate pieces. IF, however, that puzzle was thrown on the ground an infinite number of times, it seems that eventually the entire puzzle would fall in such a way as to be completely formed.

By logical extension it would seem, therefore, that one could make the argument that any sense of order within the universe is just randomness expressing itself to us as order through an ad infinitum repetition. Any order that exists in our world is just randomness manifested as order; it's just the roll of the dice wherein all five landed showing sixes. The things that we consider to have any amount of order, the human body, the life-cycle, etc., etc., could possibly just be completely random happening that appear ordered. This would mean that the word "order" would be invalid; nothing would have order, and the idea of order would be abolished in favor "things that appear ordered".

I know the rabbit-hole goes pretty deep on this one, but it's a thought-provoking idea to me nonetheless. I understand that it has to do with a function of language, word-meaning, our idea of the word "order", etc., but at the same time it's something that I like to think about. It makes sense to me, but maybe that's only because someone, somewhere, just rolled a Yahtzee.