How do you calculate %xHKO?

I'd like to go on record saying that the distribution of the sum of uniform distributions is not fun to work with

You can probably do it algebraically using multiple variable integrals of the p.d.f. (uniform distribution is P(x) = H(x) +(-1)H(x-1) ) so you end up with something like integral y = -inf to inf P(x)P(y) dy But just the fact that it's a piecewise function makes it annoying to work with.

Which, in 2 variables, just ends up being a triangle. P(x) = xH(x) - 2(x-1)H(x-1) + (x-2)H(x-2) But that limits you to finding 2hko probabilities. As a diagram, this would be like the diagonal area under a square, where the square has uniform height.

In the long run just running a few thousand simulations will probably get you close enough, and be a lot easier to write.

When it comes to two hits, you can equate this to using a diagonal line to cut diagonally across a square in the XY plane using your damage range as both the x-axis and y-axis.

ex: with a damage range of 1~7

The chance that you'll do at least X amount of damage can be calculated by finding the area of such a triangle.

7_________
6------- /
5------/
4----/
3--/
2/
1_________
1 2 3 4 5 6 7

^ It's supposed to be a 45 degree line... and a square... but whatever

The issue very slightly steps in once you cross the half way point. You'll have to work with two separate sections.


If you move this to 3 attacks, it's the equivalent of cutting a cube using a plane. And sadly this splits up the area that you have to work on into three sections. Mainly because when you begin cutting off the corner of the cube you get shapes that look like pyramids with triangular bases. (Take a cube of cheese, stand it on one of it's vertices, then cut off from the top to see what I mean.) After cutting down enough you start getting hexagonal bases. And after more you go back to triangular bases.

So now you'd have to split this into three parts to calculate the volume.

Each dimension (or hit) that you want to add into this problem will just add yet another section that you have to calculate separately, further complicating the computation if you want to be precise.




~ ~ ~ ~ ~ ~ ~

After about 25 hits though, the uniform distribution on the dice rolls/rng which used to look like this:


becomes a normal distribution like so:


This second one CAN be calculated. But only if we're talking about you taking 25 hits or more in the first place.

This would just barely a bit to calculate %4hko with something like savage blow since it hits 6 times.

I doubt most of you who use 4 hits skills would want to train on something that takes you 6 skill uses to kill. So it's pointless to look at a normal distribution.

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I'm still working on this problem. I'll let you guys know if I come up with anything.