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A Case Study in Probability: Conversion by Rarity


Yomo Kimyata

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Now, despite having used the conversion process many times, I realized that I still had a few fundamental questions as to how the process works with respect to conversion by rarity.  Now, I don’t think it is critical to know *exactly* how the process works if we can pretty easily get a more-than-working knowledge of how the process works.  A young child can learn to catch a thrown ball without knowing calculus.  But I wanted to think about the assumptions I have made, and maybe test those assumptions, because at the end of the day I’m just another curious ape.

I’ve wondered for some time how the conversion-by-rarity selects what the available pool is, and how it assigns probability of outcome.  I’ve come up with a number of possible scenarios (I’m sure there are more, so please add any you come up with!) as to how the process works.  (Of course, all this speculation would go away if someone delves into the code, but let’s indulge my idle curiosity):

  1. Determine all IOs that are available and randomly transform the IO to one from the available pool.  Example:  Starting with a level 20 Cleaving Blow IO.  There are, I believe, 195 total unique uncommon* IOs available at level 20.  This process would transform to one of 191 (not a Cleaving Blow) possible IOs.  So, roll the dice, and BAM you get a Befuddling Aura:  Confuse/Recharge.  The probability could be evenly weighted, or could have some other distribution.  If evenly weighted, we would expect each IO to come up about as often as any other.  *There are 27 “uncommon rares” which are IOs from rare sets that are listed as uncommon in the AH.  Usually, but not always, these don’t require rare salvage.  And they are not considered uncommon.  More on these later.
  2. Determine all sets that are available, then choose a set randomly.  Then choose within the set randomly.  Example:  again, starting with a level 20 Cleaving Blow.  There are, I believe, 44 unique common sets at level 20.  So, this process would choose 1 of 43 other sets randomly (again, could be evenly weighted, or some other distribution), and then choose within that set randomly (this would almost certainly be evenly weighted).  So, roll the dice and BAM you get Befuddling Aura.  Roll the dice again and BAM you get Confuse/Recharge.  If evenly weighted, we would expect to see all sets have roughly the same number of choices.  For sets with three members, we would expect to see each member come up more often than for sets with six members.
  3. Determine all categories that are available, and choose a category.  If there is more than one set in that category, then choose a set.  Then choose a member of the set.  Example:  Roll the dice and get Confuse.  Roll again and get Befuddling Aura.  Roll again and get Confuse/Recharge.
  4. Other.

So what did I do? 

PART I:  The Uncommoning

I bought 500 level 20 Cleaving Blow A/D and 500 level 20 Cleaving Blow D/E recipes on the AH, crafted them and put them in base storage.  I bought 10,000 converters under my standard budgeting rule of 10 converters per IO.  Then for the first part of the three part project, I converted each IO by uncommon rarity once and tracked the results.

Why Cleaving Blow?  I’d prefer to all start from the same IO, just in case it affects the RNG, but buying all within the same set should be almost as good.  I went with Cleaving Blow since the recipe goes from 10-50 and therefore should drop a fair amount.  I chose A/D and D/E because, well, the other two required rare salvage.

Why 1,000?  It’s a nice round number, but it’s also right on the cusp of being statistically significant.  If I could hold 2,000 IOs in my base, I might have done that, but even 1,000 items noticeably slows down processing speed in bases.  Plus, it took me longer than I liked to even accumulate 1,000 (I’d like to specifically thank the person who started ramping up bids on Cleaving Blow recipes right when I was on my last 100…)

Why level 20?  Two economic reasons:  1.  Lower is better because of crafting costs; 2.  Level 20 has enough set variety (but not as much as level 30 or 31) to get good stats, but I should still be able to sell the end results in a timely fashion.

How long did it take?  I’d say about a week or two to accumulate the recipes.  Once everything was in the bins, it took me an hour to convert all IOs and track them, following the adage that it takes five times as long to document something as it takes to actually do it.

How much did it cost?  712,671,000 inf, which was almost all invested in converters.

So what happened?  STAY TUNED!

Edited by Yomo Kimyata
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Part I:  The Uncommoning (cont.)

So, I tracked the results, then I looked at the frequency compared to what I think “normal” would look like in terms of statistics.  Here’s the data (WARNING:  STATISTICS):

 

2-standard deviations (explains ~95% of outcomes)

Spoiler
               
    By IO       By Set  
  p 0.524%       2.326%  
  p(1-p) 0.526%       2.271%  
  n 1000       1000  
  SE 0.229%       0.477%  
               
  -2 By IO 2   -2 By Set 2
  * sigma result * sigma   * sigma result * sigma
Achilles' Heel:  DD              
Adrenal Adjustment:  EM 0.06% 0.60% 0.98%   1.37% 3.10% 3.28%
Adrenal Adjustment:  EM/R 0.06% 1.40% 0.98%        
Adrenal Adjustment:  EM/R/A 0.06% 1.10% 0.98%        
Air Burst:  A/D 0.06% 0.30% 0.98%   1.37% 1.70% 3.28%
Air Burst:  D/E 0.06% 0.50% 0.98%        
Air Burst:  D/Rng 0.06% 0.20% 0.98%        
Air Burst:  D/R 0.06% 0.70% 0.98%        
Annoyance:  T 0.06% 0.90% 0.98%   1.37% 3.40% 3.28%
Annoyance:  T/R 0.06% 1.20% 0.98%        
Annoyance:  T/R/Rng 0.06% 1.30% 0.98%        
Befuddling Aura:  A/C/R 0.06% 0.40% 0.98%   1.37% 2.70% 3.28%
Befuddling Aura:  A/E 0.06% 0.30% 0.98%        
Befuddling Aura:  A/R 0.06% 0.90% 0.98%        
Befuddling Aura:  C/R 0.06% 0.40% 0.98%        
Befuddling Aura:  E/C 0.06% 0.70% 0.98%        
Bonesnap:  A/D 0.06% 1.10% 0.98%   1.37% 2.20% 3.28%
Bonesnap:  A/R 0.06% 0.50% 0.98%        
Bonesnap:  D/E 0.06% 0.60% 0.98%        
Brilliant Leadership:  A 0.06% 0.50% 0.98%   1.37% 2.90% 3.28%
Brilliant Leadership:  A/D 0.06% 0.80% 0.98%        
Brilliant Leadership:  A/D/E 0.06% 0.40% 0.98%        
Brilliant Leadership:  A/E 0.06% 0.40% 0.98%        
Brilliant Leadership:  D 0.06% 0.40% 0.98%        
Brilliant Leadership:  D/E 0.06% 0.40% 0.98%        
Bruising Blow:  A/D 0.06% 1.10% 0.98%   1.37% 2.90% 3.28%
Bruising Blow:  D/E 0.06% 1.00% 0.98%        
Bruising Blow:  D/R 0.06% 0.80% 0.98%        
Cacophony:  A/R              
Calibrated Accuracy:  A/D 0.06% 0.20% 0.98%   1.37% 2.00% 3.28%
Calibrated Accuracy:  A/D/R 0.06% 0.60% 0.98%        
Calibrated Accuracy:  A/E 0.06% 0.20% 0.98%        
Calibrated Accuracy:  A/I 0.06% 0.60% 0.98%        
Calibrated Accuracy:  A/Rng 0.06% 0.40% 0.98%        
Calibrated Accuracy:  A/R 0.06% 0.00% 0.98%        
Cleaving Blow:  A/D              
Cleaving Blow:  A/R              
Cleaving Blow:  D/E              
Cleaving Blow:  D/R              
Commanding Presence:  A/D              
Curtail Speed:  A/E 0.06% 0.70% 0.98%   1.37% 2.10% 3.28%
Curtail Speed:  A/S 0.06% 0.80% 0.98%        
Curtail Speed:  D/S 0.06% 0.20% 0.98%        
Curtail Speed:  E/R/S 0.06% 0.30% 0.98%        
Curtail Speed:  Rng/S 0.06% 0.10% 0.98%        
Debilitative Action:  A/R              
Deflated Ego:  THD              
Detonation:  A/D 0.06% 0.30% 0.98%   1.37% 2.20% 3.28%
Detonation:  A/D/E 0.06% 0.10% 0.98%        
Detonation:  D/E 0.06% 0.60% 0.98%        
Detonation:  D/E/Rng 0.06% 0.40% 0.98%        
Detonation:  D/Rng 0.06% 0.50% 0.98%        
Detonation:  D/R 0.06% 0.30% 0.98%        
Discouraging Words:  THD 0.06% 1.00% 0.98%   1.37% 2.60% 3.28%
Discouraging Words:  THD/R 0.06% 0.70% 0.98%        
Discouraging Words:  THD/R/E 0.06% 0.90% 0.98%        
Edict of the Master:  A/D              
Encouraged Accuracy:  THB 0.06% 0.70% 0.98%   1.37% 2.00% 3.28%
Encouraged Accuracy:  THB/R 0.06% 0.50% 0.98%        
Encouraged Accuracy:  THB/R/E 0.06% 0.80% 0.98%        
Energy Manipulator:  EM              
Enfeebled Operation:  A/E 0.06% 0.40% 0.98%   1.37% 2.20% 3.28%
Enfeebled Operation:  A/I 0.06% 0.30% 0.98%        
Enfeebled Operation:  A/I/R 0.06% 0.20% 0.98%        
Enfeebled Operation:  A/R 0.06% 0.30% 0.98%        
Enfeebled Operation:  E/I 0.06% 0.60% 0.98%        
Enfeebled Operation:  I/Rng 0.06% 0.40% 0.98%        
Entropic Chaos:  A/D              
Essence of Curare:  A/E 0.06% 0.40% 0.98%   1.37% 2.20% 3.28%
Essence of Curare:  A/H 0.06% 0.70% 0.98%        
Essence of Curare:  A/H/R 0.06% 0.40% 0.98%        
Essence of Curare:  A/R 0.06% 0.30% 0.98%        
Essence of Curare:  E/H 0.06% 0.30% 0.98%        
Essence of Curare:  H/Rng 0.06% 0.10% 0.98%        
Exploit Weakness:  A/D 0.06% 0.50% 0.98%   1.37% 2.60% 3.28%
Exploit Weakness:  D/E 0.06% 0.40% 0.98%        
Exploit Weakness:  D/Rng 0.06% 0.70% 0.98%        
Exploit Weakness:  D/R 0.06% 1.00% 0.98%        
Exploited Vulnerability:  DD 0.06% 1.10% 0.98%   1.37% 2.50% 3.28%
Exploited Vulnerability:  DD/R 0.06% 0.80% 0.98%        
Exploited Vulnerability:  DD/R/E 0.06% 0.60% 0.98%        
Explosive Strike:  KB/D              
Far Strike:  A/E 0.06% 0.50% 0.98%   1.37% 2.00% 3.28%
Far Strike:  D/Rng 0.06% 0.90% 0.98%        
Far Strike:  D/R 0.06% 0.60% 0.98%        
Gift of the Ancients:  D/E              
Harmonized Healing:  E 0.06% 0.50% 0.98%   1.37% 2.20% 3.28%
Harmonized Healing:  E/R 0.06% 0.10% 0.98%        
Harmonized Healing:  H/A 0.06% 0.50% 0.98%        
Harmonized Healing:  H/A/E 0.06% 0.50% 0.98%        
Harmonized Healing:  H/A/E/R 0.06% 0.00% 0.98%        
Harmonized Healing:  H/A/R 0.06% 0.60% 0.98%        
Hibernation:  A/E 0.06% 0.90% 0.98%   1.37% 3.00% 3.28%
Hibernation:  A/R 0.06% 0.70% 0.98%        
Hibernation:  A/S/R 0.06% 0.60% 0.98%        
Hibernation:  E/S 0.06% 0.30% 0.98%        
Hibernation:  S/Rng 0.06% 0.50% 0.98%        
Horror:  A/E 0.06% 0.50% 0.98%   1.37% 2.40% 3.28%
Horror:  A/F/R 0.06% 0.50% 0.98%        
Horror:  A/R 0.06% 0.70% 0.98%        
Horror:  E/F 0.06% 0.30% 0.98%        
Horror:  F/Rng 0.06% 0.40% 0.98%        
Impeded Swiftness:  A/S              
Impervious Skin:  R/E              
Impervium Armor:  R/E              
Induced Coma:  A/R              
Jaunt:  E 0.06% 0.70% 0.98%   1.37% 2.10% 3.28%
Jaunt:  E/Rng 0.06% 0.70% 0.98%        
Jaunt:  Rng 0.06% 0.70% 0.98%        
Karma:  D/E 0.06% 0.80% 0.98%   1.37% 1.60% 3.28%
Karma:  D/R 0.06% 0.40% 0.98%        
Karma:  KB Prot 0.06% 0.40% 0.98%        
Kinetic Combat:  A/D              
Kismet:  D/E              
Lethargic Repose:  A/E 0.06% 0.50% 0.98%   1.37% 2.00% 3.28%
Lethargic Repose:  A/R 0.06% 0.20% 0.98%        
Lethargic Repose:  A/S 0.06% 0.20% 0.98%        
Lethargic Repose:  A/S/R 0.06% 0.40% 0.98%        
Lethargic Repose:  E/S 0.06% 0.30% 0.98%        
Lethargic Repose:  S/Rng 0.06% 0.40% 0.98%        
Maelstrom's Fury:  A/D 0.06% 0.80% 0.98%   1.37% 2.70% 3.28%
Maelstrom's Fury:  D/E 0.06% 0.60% 0.98%        
Maelstrom's Fury:  D/E/R 0.06% 0.40% 0.98%        
Maelstrom's Fury:  D/R 0.06% 0.90% 0.98%        
Miracle:  H/A/E              
Multi-Strike:  A/D 0.06% 0.50% 0.98%   1.37% 2.00% 3.28%
Multi-Strike:  A/D/E 0.06% 0.40% 0.98%        
Multi-Strike:  A/E 0.06% 0.20% 0.98%        
Multi-Strike:  D/E 0.06% 0.50% 0.98%        
Multi-Strike:  D/E/R 0.06% 0.20% 0.98%        
Multi-Strike:  D/R 0.06% 0.20% 0.98%        
Neuronic Shutdown:  A/R              
Nightmare:  A/E 0.06% 0.60% 0.98%   1.37% 2.20% 3.28%
Nightmare:  A/F 0.06% 0.50% 0.98%        
Nightmare:  A/F/R 0.06% 0.10% 0.98%        
Nightmare:  A/R 0.06% 0.60% 0.98%        
Nightmare:  E/F 0.06% 0.00% 0.98%        
Nightmare:  F/Rng 0.06% 0.40% 0.98%        
Paralytic:  A/E 0.06% 0.50% 0.98%   1.37% 2.20% 3.28%
Paralytic:  A/H/R 0.06% 0.50% 0.98%        
Paralytic:  A/R 0.06% 0.40% 0.98%        
Paralytic:  E/H 0.06% 0.40% 0.98%        
Paralytic:  H/Rng 0.06% 0.40% 0.98%        
Perplex:  A/C/R 0.06% 0.60% 0.98%   1.37% 2.80% 3.28%
Perplex:  A/E 0.06% 0.50% 0.98%        
Perplex:  A/R 0.06% 0.50% 0.98%        
Perplex:  C/Rng 0.06% 0.20% 0.98%        
Perplex:  E/C 0.06% 0.60% 0.98%        
Perplex:  R/C 0.06% 0.40% 0.98%        
Pounding Slugfest:  A/D              
Pulverizing Fisticuffs:  A/D              
Quickfoot:  E 0.06% 0.90% 0.98%   1.37% 2.60% 3.28%
Quickfoot:  E/Run 0.06% 0.50% 0.98%        
Quickfoot:  Run 0.06% 1.20% 0.98%        
Razzle Dazzle:  A/R              
Reactive Armor:  E 0.06% 0.40% 0.98%   1.37% 2.30% 3.28%
Reactive Armor:  E/R 0.06% 0.40% 0.98%        
Reactive Armor:  R 0.06% 0.10% 0.98%        
Reactive Armor:  R/E 0.06% 0.80% 0.98%        
Reactive Armor:  R/E/R 0.06% 0.20% 0.98%        
Reactive Armor:  R/R 0.06% 0.40% 0.98%        
Rectified Reticle:  THB              
Regenerative Tissue:  H/A/E              
Rooting Grasp:  A/E 0.06% 0.40% 0.98%   1.37% 2.40% 3.28%
Rooting Grasp:  A/I/R 0.06% 0.20% 0.98%        
Rooting Grasp:  A/R 0.06% 0.60% 0.98%        
Rooting Grasp:  E/I 0.06% 0.30% 0.98%        
Rooting Grasp:  I/Rng 0.06% 0.90% 0.98%        
Rope a Dope:  A/E 0.06% 0.70% 0.98%   1.37% 2.60% 3.28%
Rope A Dope:  A/R 0.06% 0.50% 0.98%        
Rope A Dope:  A/S 0.06% 0.40% 0.98%        
Rope A Dope:  A/S/R 0.06% 0.20% 0.98%        
Rope A Dope:  E/S 0.06% 0.50% 0.98%        
Rope A Dope:  S/Rng 0.06% 0.30% 0.98%        
Salvo:  A/D              
Serendipity:  D 0.06% 0.30% 0.98%   1.37% 1.00% 3.28%
Serendipity:  D/E 0.06% 0.10% 0.98%        
Serendipity:  D/E/R 0.06% 0.00% 0.98%        
Serendipity:  D/R 0.06% 0.10% 0.98%        
Serendipity:  E 0.06% 0.20% 0.98%        
Serendipity:  E/R 0.06% 0.30% 0.98%        
Smashing Haymaker:  A/D 0.06% 0.40% 0.98%   1.37% 1.90% 3.28%
Smashing Haymaker:  D/E 0.06% 0.70% 0.98%        
Smashing Haymaker:  D/E/R 0.06% 0.50% 0.98%        
Smashing Haymaker:  D/R 0.06% 0.30% 0.98%        
Soaring:  E 0.06% 1.00% 0.98%   1.37% 2.30% 3.28%
Soaring:  E/Fly 0.06% 0.70% 0.98%        
Soaring:  Fly 0.06% 0.60% 0.98%        
Springfoot:  E 0.06% 1.00% 0.98%   1.37% 2.60% 3.28%
Springfoot:  E/Jump 0.06% 0.70% 0.98%        
Springfoot:  Jump 0.06% 0.90% 0.98%        
Stagger:  A/E 0.06% 0.30% 0.98%   1.37% 2.70% 3.28%
Stagger:  A/R 0.06% 0.60% 0.98%        
Stagger:  A/S/R 0.06% 0.40% 0.98%        
Stagger:  E/S 0.06% 0.70% 0.98%        
Stagger:  S/Rng 0.06% 0.70% 0.98%        
Steadfast Protection:  KB Prot 0.06% 0.80% 0.98%   1.37% 2.50% 3.28%
Steadfast Protection:  R/Def 0.06% 0.70% 0.98%        
Steadfast Protection:  R/E 0.06% 1.00% 0.98%        
Tempered Readiness:  A/D/S 0.06% 0.40% 0.98%   1.37% 2.40% 3.28%
Tempered Readiness:  A/E 0.06% 0.30% 0.98%        
Tempered Readiness:  A/S 0.06% 0.40% 0.98%        
Tempered Readiness:  D/S 0.06% 0.40% 0.98%        
Tempered Readiness:  E/R/S 0.06% 0.60% 0.98%        
Tempered Readiness:  Rng/S 0.06% 0.30% 0.98%        
Tempest:  A/D              
Triage:  E/R 0.06% 0.70% 0.98%   1.37% 2.30% 3.28%
Triage:  H/A/E 0.06% 0.60% 0.98%        
Triage:  H/A/E/R 0.06% 0.50% 0.98%        
Triage:  H/A/R 0.06% 0.50% 0.98%        
Triumphant Insult:  T              
Unquestioning Loyalty:  A/D 0.06% 0.60% 0.98%   1.37% 2.10% 3.28%
Unquestioning Loyalty:  A/D/E 0.06% 0.40% 0.98%        
Unquestioning Loyalty:  A/E 0.06% 0.50% 0.98%        
Unquestioning Loyalty:  D/E 0.06% 0.60% 0.98%        
Unspeakable Terror:  A/R              
Volley Fire:  A/D 0.06% 0.60% 0.98%   1.37% 1.80% 3.28%
Volley Fire:  D/E 0.06% 0.40% 0.98%        
Volley Fire:  D/R 0.06% 0.80% 0.98%        
Volley of Velocity:  KB/A 0.06% 0.80% 0.98%   1.37% 2.00% 3.28%
Volley of Velocity:  KB/D 0.06% 0.50% 0.98%        
Volley of Velocity:  KB/R 0.06% 0.70% 0.98%        

 

3-standard deviations (explains ~99% of outcomes)

Spoiler
               
    By IO       By Set  
  p 0.524%       2.326%  
  p(1-p) 0.526%       2.271%  
  n 1000       1000  
  SE 0.229%       0.477%  
               
  -3 By IO 3   -3 By Set 3
  * sigma result * sigma   * sigma result * sigma
Achilles' Heel:  DD              
Adrenal Adjustment:  EM -0.16% 0.60% 1.21%   0.90% 3.10% 3.76%
Adrenal Adjustment:  EM/R -0.16% 1.40% 1.21%        
Adrenal Adjustment:  EM/R/A -0.16% 1.10% 1.21%        
Air Burst:  A/D -0.16% 0.30% 1.21%   0.90% 1.70% 3.76%
Air Burst:  D/E -0.16% 0.50% 1.21%        
Air Burst:  D/Rng -0.16% 0.20% 1.21%        
Air Burst:  D/R -0.16% 0.70% 1.21%        
Annoyance:  T -0.16% 0.90% 1.21%   0.90% 3.40% 3.76%
Annoyance:  T/R -0.16% 1.20% 1.21%        
Annoyance:  T/R/Rng -0.16% 1.30% 1.21%        
Befuddling Aura:  A/C/R -0.16% 0.40% 1.21%   0.90% 2.70% 3.76%
Befuddling Aura:  A/E -0.16% 0.30% 1.21%        
Befuddling Aura:  A/R -0.16% 0.90% 1.21%        
Befuddling Aura:  C/R -0.16% 0.40% 1.21%        
Befuddling Aura:  E/C -0.16% 0.70% 1.21%        
Bonesnap:  A/D -0.16% 1.10% 1.21%   0.90% 2.20% 3.76%
Bonesnap:  A/R -0.16% 0.50% 1.21%        
Bonesnap:  D/E -0.16% 0.60% 1.21%        
Brilliant Leadership:  A -0.16% 0.50% 1.21%   0.90% 2.90% 3.76%
Brilliant Leadership:  A/D -0.16% 0.80% 1.21%        
Brilliant Leadership:  A/D/E -0.16% 0.40% 1.21%        
Brilliant Leadership:  A/E -0.16% 0.40% 1.21%        
Brilliant Leadership:  D -0.16% 0.40% 1.21%        
Brilliant Leadership:  D/E -0.16% 0.40% 1.21%        
Bruising Blow:  A/D -0.16% 1.10% 1.21%   0.90% 2.90% 3.76%
Bruising Blow:  D/E -0.16% 1.00% 1.21%        
Bruising Blow:  D/R -0.16% 0.80% 1.21%        
Cacophony:  A/R              
Calibrated Accuracy:  A/D -0.16% 0.20% 1.21%   0.90% 2.00% 3.76%
Calibrated Accuracy:  A/D/R -0.16% 0.60% 1.21%        
Calibrated Accuracy:  A/E -0.16% 0.20% 1.21%        
Calibrated Accuracy:  A/I -0.16% 0.60% 1.21%        
Calibrated Accuracy:  A/Rng -0.16% 0.40% 1.21%        
Calibrated Accuracy:  A/R -0.16% 0.00% 1.21%        
Cleaving Blow:  A/D              
Cleaving Blow:  A/R              
Cleaving Blow:  D/E              
Cleaving Blow:  D/R              
Commanding Presence:  A/D              
Curtail Speed:  A/E -0.16% 0.70% 1.21%   0.90% 2.10%