I'm not a "mathematician" by trade, I'm a software engineer, so there may be a better way out there, but I figured I'd write up what I do know about just in case my perspective was useful.
The best ways I've found for understanding these kinds of probability problems is with a dedicated calculator designed for them.
So, for example, you can use [Anydice with a "program"](https://anydice.com/program/13e) saying
output [highest 3 of 4d6]
for your initial example, and it provides a graph showing the distribution. You can change the "3", "4", and "6" for any values of *k*, *n*, and *s* you want.
My understanding (which may be wrong, but *think* is how it works) is that Anydice isn't a numeric Monte Carlo approximation or the like, but that it actually works through all the possibilities for each die being rolled to find the real probability curve. I'm not aware of the source code being publicly available, though, but even if it were available I'm not sure one could derive a general formula from it for all possible *k*/*n*/*s*.
In general randomness has been tricky to model and easily see the patterns of, even with modern programming languages. As "big data" analysis grows as a field, even more tools designed to help model probabilistic events are being developed like "[Bean Machine](https://ericlippert.com/2020/09/23/introducing-bean-machine/)". You might be able to use tools like those to look for the "general insight" you're searching for.
Peter Cooper Jr.
·
2020-10-25T02:56:32Z (about 4 years ago)