Degrees of Success

Hey there folks! Aidan here.
I'm was doing some preliminary research for a TTRPG project and thought it'd be cool to talk about dice probability, and how different systems use different approaches to get similar results. This is going to get a bit mathy, but I'll try to keep things interesting.

Dice Resolution Mechanics

Before we start talking statistics and probability, lets set a baseline for what I mean by Dice resolution. Typically in a tabletop roleplaying game, a user rolls dice and, depending on their stats and the difficulty of the check, their either succeed or fail. Some basic examples of this are:

• Roll a d20, add a modifier, if it's over a number, you succeed and complete your task!
• Roll one or more dice, for each die that rolls over some number, you get a success! If enough dice succeed, you complete your task!
• Roll a d100, if it's under your stat, you succeed and complete your task!

There are, of course, variations on these and more abstract versions, but this is a generalized group of the ones I've seen most often. Systems that use these are often reffered to as "Roll Over", "Die Pool", and "Roll under" Systems.