I think you guys are waaaaaay off.
You're discounting the chances of duplicates in truly random packs, a variation of the birthday problem. Also you're ignoring the R+, which Decipher claims was "33% more rare than rare." It's funny, they say that Mirror, Mirror included "a handful of Rare+ cards (33% more rare than rare), and a distribution of Rare, Uncommon and Common cards, similar to other Star Trek CCG sets." And by "a handful," they meant "more than the number of rares." There are 30 R+ cards and 20 R cards. Some handful!
I don't know what the exact distribution was, but people throw around 121 cards on a sheet as a number. If the rare sheet had them all, and if there's 1 UR on the sheet, 60 R+s (2 copies of each) and 60 Rs (3 copies of each), then you're chances of completing a full set are about 0.0001% in 3 boxes (30 packs a box is 90 packs). If you don't care about the UR, then your chances climb to 0.0005%.
I could be wrong about the R+ distribution on the sheets. Let's be generous and say there are 50 rares, all equally likely to be in a pack, no R+s, and we don't care about an Ultra-Rare. Then your chances of getting a complete set rise to about 0.0009%, with 3 boxes or 90 packs. To have a 90% chance of getting a full set of 50 rares, you'd need to open around 300 packs, which is 10 boxes. That's the birthday problem for you.
All of this is assuming that commons and uncommons aren't an issue of course.
I'm also assuming that the packs are truly random, rather than being correlated. In other words, if two neighboring packs in the same box are likelier to have their rare cards have also been neighbors on the same print sheet, then the math goes out the window entirely. It depends on whether or not Decipher shuffled the cards or shuffled the packs before packing the boxes.
I know this assumption was not the case in at least one high profile case for one game (Magic?), where people could figure out which pack in the box to purchase from the store to pick out the super-valuable rare, if they knew what one of the other packs already contained, based on that pack's location.