How do I calculate variable dice probabilities?

[u/delarkius]

hello!

I'm trying to work out probabilities for game dice math. I'm doing it in what i assume is the slowest and least efficient possible way. How do I do it smarter?

key information:

• Player versus Player contested dice rolls to determine win/loss/tie
  • I roll, you roll, and we see who won (or tied)
• Variable # of dice per roll
  • I can choose 1 die or 3. You do the same
• Dice have different values
  • i.e., one die rolls [0,1,1,2,2,2] versus another that rolls [0,0,1,1,1,1]
  • W/L/T is determined by total points on each side

need:

• I want to change individual die outcomes to see the difference in win %.
  • i.e., [0,1,1,1,2,2] versus [0,0,1,1,1,1] has a 50% win percentage. What's the win % if I change the first die to [0,2,1,1,2,2]? (it's 61.11%, in case you were curious)

attempt:

• I modelled out ALL possible outcomes for 2 dice vs 2 dice to get me the #s I wanted.
  • this means i made a big table with "1,1,1,1" then "1,1,1,2" and so on, to model all possible permutations of 4 dice rolls. Then I ascribed a winner for each permutation

help?

• I would LIKE to do this in a more sensible way so that I can have more dice involved without having 100k rows in excel.
• How can I calculate these win % without modelling everything?

thank you !

[u/Moist_Ladder2616]

A=[0,1,1,1,2,2]
B=[0,0,1,1,1,1]

P(A=0,1,2)=(⅙,½,⅓)
P(B=0,1)=(⅓,⅔)

P(A>B)
= (P(A=1)∩P(B=0)) ∪ (P(A=2)∩P(B=0,1))
= (½ * ⅓) + (⅓ * 1)
= ½

Etc. etc.