So have you been trying to get a super rare (or super common) item and haven't been getting it? If it's the super rare item, you should probably keep trying. But if it's a super common item...maybe you're not quite getting it right with increasing your chances. And that's what this article is for, to provide a basic explanation of how the game hands you your quest rewards.
First before we discuss quest rewards, make sure you're carving the darned monster's corpse . Seriously! Each monster gives you 3 carves, more if they're bigger (Gammoth for example gives you 4 carves) and Elder Dragons often offer more carves than 3 as well. Severed tails give you 1 carve, 2 if it's Deviljho's fat pickle tail. Carve rewards are really straightforward. You carve, and RNG decides what item you get from a fixed pool of parts you can obtain. Not all of a monster's parts can be obtained from carving. Shells and scales are going to be the most common, with plates and rubies/other gems being frustratingly rare.
If you want to increase the number of carves you get out of bodies and tails, Carving Celebrity can help. Carving Celebrity increases the number of carves you can get out of anything by one. Note that it won't stack with the kitchen skill Felyne Carver (lo or hi).
Now that that's out of the way, let's talk about that reward box! The quest reward box can be broken into two colors: Red and Blue . The red rows are for your standard, run-of-the-mill quest rewards. You get them for completing the quest. No extra work required! The top row is the "A" rewards and usually consist of monster parts. The bottom row is "B" rewards and usually consists of miscellaneous other items. You're guaranteed at three tiles of items in each row, with RNG being the judge of whether you get more. The armor skills Great Luck and Good Luck will greatly increase your chances of getting lots of quest rewards. You can also eat in the kitchen for Lucky Cat or Ultra Lucky Cat but note that it won't stack. Also, Crazy Lucky Cat only increases zenny.
Blue rows will represent the rewards you get from breaking or wounding parts of a monster, or if you capture the monster (since capturing nets you no carves aside from a possible tail carve). For every part you break, RNG determines what reward you get in the blue rows of the reward box, if any. Upon capturing the monster, you will get guaranteed two items in the blue box that will be monster parts, and RNG determines if you get a third one. For a chance at a fourth reward, the armor skill Capture Expert or Capture Master will help you with that.
So what about subquests? Well, they function similarly, with their own reward pools. You just get fewer rewards since the subquests are usually not as difficult as actually killing the monster itself, and if the subquest is something silly like delivering account items or killing small minions, the rewards probably aren't anything worth writing home for.
So how are you supposed to apply this knowledge? Well now that you know where the rewards are coming from, it gets easier to optimize fast farming runs if you're after rare items like plates or rubies, etc. If you are prioritizing speed of farming over optimization of probability, running in and cutting the tail/causing the monster to drop a shiny could be all you need to be fast and get that rare item eventually. Or, if you're optimizing your chances, breaking all the parts, popping off the tail (if possible), dropping shinies and picking them up, and then carving or capturing will maximize your chances.
Or, sometimes, the subquests involve breaking a specific monster's part. Doing the subquest over and over is fast and has a decent chance of giving you what you need.
If you're not really into that, and you're still wondering what the point of this section is, here's why this is important: You can speed through the quest and not break any of the monster's parts, but your rewards will suffer! And furthermore, many items like horns and tails are not available unless you break a monster's part! So take it slow and make sure you've got all the wounds on the monsters where you want them before closing in for the kill. Oh, and capturing won't get you less rewards than carving. They even out in the end.