I am trying to create a metric to sort alchemy ingredients by in Skyrim, I wanted to sort the ingredients by "probability of yielding a potion with the number of ingredients I have". The way I was looking at the problem is that it is a combination of the Birthday paradox and a binomial trial, but I'm having trouble mixing the two concepts, because 1) it's hard 2) I never took a proper stat class. I'm a physicist by training, and my stat education was all based on error propagation and curve fitting.
If you haven't played Skyrim, these are the rules (R) for the alchemy mini-game:
- Skyrim has ingredients which are combined into potions.
- Each ingredient has 4 effects, which must be "discovered."
- You can attempt to make a potion with 2 or 3 ingredients. If any of the effects in the combination are the same, a potion will be created with those matched effects.
- A potion can be created whether the effects were known or unknown before the combination.
- When you combine 2 ingredients, whose effects were unknown before the combination, and those ingredients both have the same effect, the effect will be discovered for both ingredients.
- The game keeps track of failed combinations of ingredients. If you select ingredient I, ingredients I' that produced a failed experiment will be greyed out. It is currently unknown if this is "pullable data", but it is immediately obvious when playing the alchemy mini-game.
- A Skyrim alchemy tutorial, with relevant part starting at 8:28 https://www.youtube.com/watch?v=3E3PkxusHDc&t=508s
- The end product would hopefully be a mod that would sort the alchemy ingredients by the metric. It would either be a modification to, or require the mod called SkyUI. See: https://www.nexusmods.com/skyrimspecialedition/mods/12604?tab=posts
Assumptions (A) for modelling the problem
- Total number of effects is known and fixed (56, e.g. Ravage Health, Fortify Barter, Restore Magika)
- Potions are created using 3 ingredients
- When searching for the effects ingredient I, ingredients I' and I'' will be chosen so that no common effects are known. i.e. Skyrim reports "A potion of unknown effect."
- For a given ingredient, all 4 effects are different. See: https://elderscrolls.fandom.com/wiki/Ingredients_(Skyrim)
- No two ingredients have all 4 of the same effects.
- The number of ingredients can change based on DLC's and mods to the game, but is approximately 80-200.
- The game's "sell value" of an ingredient is a good proxy for its rarity.
- Not that important at this stage, but if I am to implement this into a mod for Skyrim, I have these functions to work with (I can create more, or make lookup tables): https://www.creationkit.com/index.php?title=Math_Script
- For any other imposed assumptions or approximations, an error or uncertainty of <=1% is negligible, and up to 10% is acceptable. E.g. an approximation of a binomial coefficient, swapping a binomial distribution for a gaussian, hack to decrease calculation time.
Conjectures (C) (statements that are probably true enough to model the problem with)
- If you are trying to determine unknown effect(s) of ingredient I, it doesn't matter if the effects of I' or I'' are known or not.
- Effects are randomly and uniformly distributed amongst ingredients, given conditions A4, A5. Not true, but close enough, see https://elderscrolls.fandom.com/wiki/Ingredients_(Skyrim).
- A3 implies that each experiment is testing the unknown effects of ingredient I, amongst 8 unique effects in ingredients I' and I''. This isn't completely true, since if you make a combination of 3 ingredients where I' and I'' have at least 1 unknown effect, it is possible that you could discover a matched effect between I' and I'', but not discover an effect in I.
Hypotheses (H)
- A given experiment/combination/trial, can be modelled as a birthday paradox. With a given unknown effect checking for matches amongst 8 effects in ingredients I' and I''. With the number of people in the room being subbed for number of effects (8? 9? 10? 11? 12?), and days of the year being subbed for the total number of effects (56)
- Repeated trials can be modelled with the binomial distribution.
- The probability of a match from the birthday trial gets plugged into the probability of success in the binomial distribution.
Questions (Q) for the Internet
- Are hypotheses 1-3 correct?
- How do I model the birthday problem? It's like 4 people go into a room of 8, none of the 4 entering people have the same birthdate, and the people in the room could have 5-8 unique birthdays selected from 56 days.
- For the binomial model, I need "at least 1 success", how do I model that? Is it using the (binomial?) distribution equation, or the formula for the definite integral of the distribution?
- Is the chance of a creating a potion based more on the first ingredient, or the 2nd and 3rd ingredients?
- What are some resources that would help me solve this problem? Assume that I have access to the internet, LibreOffice, scihub, torrent sites, but not a library or a college math department. I theoretically have access to Mathematica, but it would take 1-8 hours to set up on my Raspberry Pi 3B (I tried yesterday, and I think I FUBARed the OS, hopefully not the Pi or SanDisk).
Equations (E) I would actually like to obtain/solve using my mod/code/equations.
For the following, Let: I be the ingredient under test, k = number of known effects of ingredient I, T=total number of effects in Skyrim (56), q=quantity of ingredient I (practical range 0-10000), n=number of ingredients in inventory (minimum 3, no theoretical max, practical range 40-100).
- Given 3 Ingredients, where none of the effects are known, what is the probability that a potion will be created in 1 trial.
- Given an ingredient I, with k of 4 known effects, what is the probability of of creating a potion with ingredients I' and I'', assuming Skyrim says "A potion of unknown effect"
- What is the probability of discovering effect I_e (e is a subscript from 1-4), amongst all ingredients, within n trials, where n is the quantity of ingredient I in inventory.
- What is the expected gold wasted (game value of ingredients) before effect I_e is discovered, performing the search required for Equation 3.
