Pfletch, I know you are a smart guy and probably know a heck of lot more than me about hd or tactical shotguns, but you are way out of line with your suggestion that I don't get anything or don't want to admit it.
If you simply read the question, I ask which is
more likely to put an instant stop the threat if your hit is 8" to left or right of center at a distance of 10 yards?
I wrote out a lengthy explanation of the odds of placing pellet in specific areas of the person, but decided not to post it. You probably could follow it with ease, but I would end up having to explain every little nuance to many mall ninjas (a category in which I do not include you and a few of the others ).
So I simplified greatly.
At 10 yards, which is a long shot for HD but that was the question, I expect a similar size pattern for a 12 or .410 short barrel open cyllinder. The .410 000 3" magnum has 5 pieces of shot. The 12 #4 3" has 41. Both are Remmi express.
With the pattern of each centered 8" to the left of center, I believe you are likely to get 29 hits with #4 and 12 misses. With the 5 pieces of 000 in the .410, you may only get 3 or 4 hits.
Notice my question asked about to the right or left of center. If you off to one side, you are likely to place much of the pattern over the heart and may put some pellets into the left ventricle and the apex, which likelihood increases if you have more pellets. The heart doesn't lay that deep and there is little or no need to penetrate more than a few inches. The other side is a different matter. Still, with 8 times as many bits of shot, the #4 is more likely to place more lead nearer to center mass.
Regardless, the focus of the question was which is most likely
and not if either was likely to stop a BG. I submit the 12 loaded with #4 is the most likely.
I also asked about risks of collateral damage
. No one addressed the issue since they know it is much greater from 000 that #4 due to the fact that the 000 given the same velocity much more likely to penetrate walls and even the BG and hit innocents.