Q10 Anyone have a probability distribution for a modified opposed d20 check, either in chart or formula form? Like, what's the probability of winning initiative if your initiative is seven greater than the opponent's?
well, what you are looking for is a multiple-variable regression (which involves differential equations) -- which is just a pain in the ass to do (well, for me, it's just more work than I'm willing to do for my D&D).
if anyone here is an actual mathematician or statistician, then they might be more inclined to help. (I'm not trying to be a smart-ass or anything ; just trying to be realistic -- FWIW).
I took differential equations, but I've already forgotten pretty much everything about that course... though for that much work, I could just brute-force the scenario more easily. Ah, well. Looks like I'll be doing the math myself before too long.
The math is pretty simple if you think about it as a process rather than, well, math. You've got X chances in 400 that you win, Y chances in 400 that you lose, and Z chances in 400 that you start the process over again without any changes. If you ignore the part that just repeats, because it returns you to the original probability distribution, then the math simplifies to the following:
S = (190 + sigma[n=(0,D),(20-n)])/(400-20+D)
S is the chance of success for the person with the higher modifier.
D is the difference between the higher and lower modifiers.
I'd explain the sigma part properly math-like, but I can't seem to think of it other than as a for loop.
for (int n=0; n < D; n++)
sigma += (20-n);
Anyways, this should properly model the actual distribution. If you want to directly calculate the chance of losing (or of the lower modifier winning), just subtract the sigma part in the formula rather than adding it.
Some common values:
D+0: S = 190/380 = 0.50
D+1: S = 210/381 = 0.5512
D+2: S = 229/382 = 0.5995
D+3: S = 247/383 = 0.6449
D+4: S = 264/384 = 0.6875
D+5: S = 280/385 = 0.7273
D+10: S = 345/390 = 0.8846
D+15: S = 385/395 = 0.9747
D+19: S = 399/399 = 1.00
D+20: S = 400/400 = 1.00