In an ordered world, what is random must be so by design. You have to "know" it is random.
Now it seems relevant what St Pope John Paul II said:
"All the observations concerning the development of life lead to a similar conclusion. The evolution of living beings, of which science seeks to determine the stages and to discern the mechanism, presents an internal finality which arouses admiration. This finality which directs beings in a direction for which they are not responsible or in charge, obliges one to suppose a Mind which is its inventor, its creator."
So suppose you find something that satisfres your criterioni of randomness, dies that mean that God does not have a reason for allowing it, has a way it fits in non-randomly with a higher order?
We really need a lower-level idea of what 'random; would mean to you. It cannot mean something that excapes the reason of God.