Despite one of the staff members commenting, I'm afraid my experience with Genesis tells me a completely different tale to how it works. From observing how the genesis drops went, this is how it seems to function in reality:
At first your only drops are shards and cloaks. You keep getting blue bags and its shard, cloak, shard, cloak over and over again. Until you get about 25 of them, then you will get a chance for a new type of genesis, which is indeed determined randomly. You may not get it instantly and go on to about 30 or 40 blue bags until you get this priveliege. After that you get, lets say, an aspect.
Once you aquire your new type of genesis, in this case an aspect, you will start seeing aspects rain down on you over and over, while shards and cloaks get more rare. After about another 20 or 40 aspects, you will get a different type added to your loot roster. In my case that was fragment. Then you will start seeing a ton of those as well.
I have yet to get a remnant. I don't remember off the top of my head, but I believe I'm over a hundred bags in. Maybe 110ish, or more.
You could write it off as simply the luck of the draw, but it seems to have a very defined pattern from what I've noticed, and I'm inclined to believe that rng plays a smaller part in this system.
Giladar - rr 80 DPS AM