"Show Your Work" the 3d4 vs 2d6 follow-up

Hey all,

This Saturday, I posted a video to the All Funnies and Games Youtube channel. If you haven't seen it already, you can watch the video here.

What's there is enough to answer our question, which was, "If you can select either 2d6 or 3d4 for a best two out of three roll-off, and leave your opponent with the other, which should you pick?" However, that's not all of the supporting number-crunching necessary to be sure of the result. (The result was 3d4, by the way). So, here, I'll be posting a chunk of dice mechanics-related content to, hopefully, give you some food for thought in looking deeper into these mechanics yourself.

Defining the Formula
- We will write our formula as xdy.
- x = the number of random number generators whose outcomes will be added together
- y = the highest possible outcome of each random number generator used, these random number generators can only produce a whole number between 1 and y
- each random number generator will produce one outcome, all outcomes will be added together, the sum of these will be called the "result" of xdy

Rules
Rule 1: The lowest possible result of xdy will always be x

Rule 2: The highest possible result of xdy will always be x times y

Rule 3: The number of possible die results will always be (x times y) - (x-1)
--For the purposes of rule 3, results does not differentiate a 7 made from a 2 and a 5, from a 7 made from a 1 and a 6, etc.

Rule 4: The number of possible die results made from different combinations of die faces will always be y^x
--For the purposes of rule 4, results DOES differentiate a 7 made from a 2 and a 5, from a 7 made from a 1 and a 6, etc.

Rule 5: The likelihood of getting either the most or least likely result is always 1:y^x

Rule 6: If (x times y) - (x-1) results in an even number, the most likely result will NOT be a whole number.

Rule 7: IF (x times y) - (x-1) results in an odd number, the most likely result will be a whole number.

Rule 8 [(y+1)/2] times x = average result of xdy.

Tables

Here's hoping this is some information that'll come in handy as you look into dice and statistics yourself. What resources have helped you plan probabilities in your own tabletop games? Let's talk about that in the comments. Happy Gaming, all.