Hi everyone!
So, I'm very familiar with looking at the statistics of d20 rolls from the perspective of average rolls (10.5) and against static DCs, but how do the odds change for opposed rolls? What if the amount of a success or failure mattered; how do the statistics change then?
Xeviat-DM wrote:
i posted the results in one thread... forgot which one it was now.
Which thread?
Say it's a single opposed roll, with the "attacker" winning ties. Set X = attacker's bonus - defender's bonus. Then the probability for the attacker to win is
(20+X)*(21+X)/800 if X is negative (from -20 to 0)
and
(20+X)*(21+X)/800 - X*(X+1)/400 if X is positive (from 0 to +19)
If that helps. Hope I did it right 
Well, I just remembered I did something like this a while ago for the Ring of Anticipation in 3.5, and I was in the process of prettying it up for 5e - I did it brute-force though, since back then I was using it in an argument and I was so far along that this actually took less time than the derivation would. The result's here (I need to find better ways to procrastinate). It's named "initiative" because it was intended to handle rings of anticipation, but it works for any contest.
This version includes ties as well as advantage and disadvantage. I denote that on tabs with + and -, respectively; it's always set up to display the results as the chance that Alice will beat Bob. "Swing" is the difference in relative modifiers; because the percentages are for Alice to win, positive means in Alice's favor, negative means Bob's favor.
The interesting thing to me was that the results were identical in a few places: advantage on Alice is identical to disadvantage on Bob, and vice versa. That cut down quite a bit on repetition. You'll also note that advantage and disadvantage work seriously in your favor, considering how many ways you have to impose them - for instance, if Alice is a 4th level fighter with 14 Strength and Bob is a 20th level fighter with 20 Strength, both trained in Athletics, Alice sees a -7 swing. A straight-up Strength (Athletics) contest between them is almost a foregone conclusion (19.5% chance for Alice to win, 77.3% chance for Bob to win). But if Alice gets advantage and Bob gets disadvantage, those odds shift to 48% Alice and 46.6% Bob - which, for the record, is comparable to an even contest (no advantage/disadvantage, equal Athletics modifiers).
Jaelis' formulas are slightly off - it actually gives accurate values for "X+1". (That is, if X=0, it gives you accurate values for when you have a +1 modifier relative to your opponent, and so on. This is true for both formulas.) This works for cases where there's no advantage or disadvantage - each individual outcome is equally likely. That distribution needs to be changed to the distributions for assorted combinations of advantage or disadvantage.
Tempest_Stormwind wrote:
jaelis wrote:
My thought was that most of the time one of the parties will "win" a tie for one reason or another, so I was calling that character the attacker.
But I suppose you're right that in some cases a tie is a distinct third outcome; then of course you'd have to pull that out separately. At least the chance of a tie is relatively easy to calculate.
I'm more worried about odds for the initiator winning by certain amounts, as I'm working on some ideas that involve opposed rolls with degrees of success.