If you have been doing the Anima Weapon stage 1 where you need to grind FATE all over Heavensward maps for your Luminous Crystals, you will realize that you will get your first crystal of that particular map very quickly, second one will take a few FATEs and the third one will take quite a while. This has led me to believe that the drop rate of the first crystal is probably quite high (maybe 80%?) and it drops off to around 10% for the third one.
With this theory in mind, whenever I get a crystal from a FATE, I would entrust it to my retainer, hence the game "sees" that I do not own any crystal and the drop rate for the next crystal will be at 80% (or whatever the droprate for the first crystal is).
Long story short, whenever you get a luminous crystal from a FATE, entrust it to your retainer and continue farming for the next, you will get it much sooner. I just got all 3 crystals in Azys Lla in 3 FATEs! Good luck!
Check out the link and look at the chat log
http://imgur.com/mGV8gIr