To use the analogy-
Your method is 3d6 in order, discard all generated characters with an ability score below 14.
Round up all the men in the world, shoot all men under 6' 3".
In both cases, you will have an output (the finally generated characters, the men still standing) that is different because of the alteration. But the results are based on the original pool, skewed by the effect of the discard.
So what does that all mean? Well, for starters, you're going to have a lot of discards if you're using 3d6 in order, discard all characters that have an ability below 14. And also a population well-suited to basketball.
(This may be obvious, but it is meaningful.)
If you only generate results "in full," sure, it'll take a long-donkey time--P(14+ on all 6 rolls) = 0.162^6 = 0.0000181, approx, or 1 in every 55,324 rolls. But I would think most people, if trying to apply these limitations, would not do that, because they'd know better. They'd roll a number of times, perhaps all 6 if you have 18d6 lying around, in clearly separate sets of 3. If you then find that any of those sets are below 14, you reroll
only those triplets that are below 14, until they're not. That's
far less cumbersome than rerolling all 18 dice until you luck into a set of all-14-plus. And, you're in luck--the expected value is around 1 dice-triplet of 14+ per set of 6 stats, so you
should shave off at least one roll, on average, every time you try. Plus, the chance to get multiples compounds the speed-up, while the chance to get none at all only slightly slows things down (because getting no stats at all that are 14+ is low-probability: 1-(1-.162)^6 = .654, so about two thirds of all initial sets should have *at least* one, and more than half, p=0.587, of five-triplet sets should get at least one 14+, etc.)
There's more you can do, though. Frex, if 14 is the cutoff, you can simplify the process a bit more by excluding 1s from the distribution (it's not possible to have a roll of 14 that includes any number of 1s; it is
just barely possible to achieve 14 with a single 2 e.g. {2,6,6}), and this can be done without needing "exclusion rules" by doing 3d5+3. This gets the probability of 14+ up to a nice, round 0.28, so getting all six at once = 0.28^6 = 0.000482, or approximately 1 in 2075 sets. You've got a decent chance of getting at least two stats of 14+ on the very first roll, meaning you'd only need to reroll 4 sets, making things dramatically faster.
And, of course, you can still keep the six results you got 'in order' or not--whether you choose one or the other is *truly* unrelated, as long as it's acceptable to discard only those parts of a set that disqualify until they qualify.
Also: this post has been published posthumously, as I was three inches too short. Alas!
