So, every so often, someone asks what the probability is of roll a perfect 108 statline. Consistently, people give a lot of wrong answers (1/6^18 is a common favorite), and a lot of "I don't know"s. Because, well, the problem is hard. The game's built-in rerolling of too-low stats complicates the probabilities dramatically. For a long time, all we knew was that the true probability was very low, but much higher than 1/6^18. Until now.
Through the wonders of linear algebra and Matlab, it's now possible to instantly compute the probability of rolling a 108 (or any other total) for any given race/class combination. Multiclasses are still a work in progress, but that's just because I'm excited to post this, not because they're at all hard to add.
So what are the odds? Well, there are a lot of possible combinations, so I'll just post the ones that give the best and worst odds:
Human fighter (/thief/cleric/mage): 1 in 496.25 billion
Human paladin: 1 in 5.3104 billion
Elf ranger: 1 in 198.03 million
Note that I made a couple assumptions when computing these probabilities. In particular, I assumed that the game rerolls any statline below a given minimum (which is supported by my testing), and that racial modifiers (the elf +1 Dex, for example) are added after the roll is made but before the roll is compared to the various minimums (this is less supported by evidence, but I'm not sure how else they'd do it).
EDIT: I'm also assuming the game rolls 3d6 to generate stats, as opposed to 4d6 drop lowest. This is most likely correct, but the two are very difficult to distinguish without extensive testing.