Then what does "random" mean in your world?
It isn't in "my world", it's what the word means. This is about a statistical argument. Statistically speaking, "random" means it has a uniform distribution, in that any possible outcome is equally likely. That's it. It has nothing to do with a lack of intelligent bias.
In more casual usage people use "random" to mean "by chance". Even though random effects play a role, evolution is NOT a random process.
Randomness would only be relevent to DNA formation if the it was postulated that an entire DNA molecule formed at one time, out of its component parts. Nobody believes that. The randomness argument is not relevent.
"Random" simply means unaided by any intelligent bias.--Post 81
From www.dictionary.com:
random
Mathematics & Statistics. Of or relating to a type of circumstance or event that is described by a probability distribution.
There are natural occurances that are non-random. Chemical reactions are non random, for example.