Diablo® III

On the Probability of Rolling Trifecta on Craft

TL;DR

Here is the graph for the TRIFECTA + AR:
http://i1306.photobucket.com/albums/s565/drag0re/TRIAR_zps09a5180e.jpg

Here is the graph for the TRIFECTA ALONE:
http://i1306.photobucket.com/albums/s565/drag0re/trifectaalone_zps8ea94311.jpg

Y axis: probability to craft a trifecta...
X axis: ... after that many attempts.


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After reading some of the archon crafting show off threads I have come to wonder how many trials it would take to eventually craft a pair of insanely godly gloves. I found similar attempts on a number of different threads, but there the models were oversimplified and rather dubious (e.g. some dude was trying to adapt the statistics of poker directly to stats roll, but left out many key assumptions that needed to be taken into account).

So I'm going to work out the math for the Archon Gauntlets of Intelligence as an illustration. I might do the same thing for the amulet / shoulder at some point.

Let's first look at the most ideal (and easiest to compute) combination of stats, regardless of their values:

    +Inherent Main Stat (Int)
    +Int & Vit
    +AR
    +AS
    +CC
    +CD


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THE ASSUMPTIONS

  • There are affixes that add to a single main stat, and affixes that add to two main stats at once (double stat roll, max +100 for each).
  • It’s impossible to roll a +mainstat that’s the same as a single inherent mainstat bonus EXCEPT if this +mainstat comes from a double stat roll.
  • Mainstats (other than inherent) can't roll more than once EXCEPT if this +mainstat comes in part from a double stat roll and a single stat roll.
  • This does not preclude double and single stat rolls from occuring twice, so long as the above condition is still satisfied.
  • All properties have an equal probability to roll, among a pool of 25 possible stats for rare gloves; in other words, the probability that your first stat rolls +strength is the same as the probability that it rolls +pickup radius.

    • THIS ASSUMPTION MIGHT BE FLAWED, as I've read several threads were the OP reported the occurences of main stats vs. other type of modifiers on a sample of 100 crafted items, and some mainstats seemed to have a slightly better chance to roll than others, in addition to the fact that mainstats also seemed to have a higher chance to roll than other modifiers. So for now the following resuls are be taken as an upper bound for the number of trials it would take on average until you can get the desired roll.


  • In addition, these events are INDEPENDENT (again getting +pickup radius on the first modifier will not affect the probability of getting +dexterity on the second roll). Of course this is not true if say +Int has already rolled twice total. The probability of getting another +Int roll would then be 0.
  • source:
    http://diablo.incgamers.com/blog/comments/diablo-3-crafting-strategy-the-new-archon-items
    http://diablonut.incgamers.com/affixes/?c=adventure-and-stats&l=1&u=70&h=119253633
    http://www.diablofans.com/topic/41045-spoiler-diablo-iii-item-affixes/#entry863934
    http://us.battle.net/d3/en/forum/topic/7923992596

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    THE MATH

    So under the assumption of independence I can proceed as follows: I consider the rolls one by one, having no influence on each other except when: 1) stat already rolled in one or two of the previous rolls, in which case it cannot roll a third time.

    MAIN ROLL
    +[201-230] Intelligence.
    This one is guaranteed.

    FISRT ROLL
    Let's calculate the number of combinations we can get. We technically have 24 stats to choose from (25 minus the inherent stat, here Intelligence). But we also have to take into account all the possible double stat rolls: Int/Vit; Int/Str; Int/Dex; Str/Vit; Str/Dex; Dex/Vit. That gives us 6 pairs of stats. So we have a total of 24 + 6 = 30 events, each with a probability of 1/30 to occur. So let's say that +Attack Speed rolls first.

    SECOND ROLL
    So we're left with 23 stats to choose from (24 minus AS), in addition to the remaining 6 pairs of double rolling stats. We thus have a total of 23 + 6 = 29 events, each with a probability of 1/29 to occur. So suppose +Critical Chance rolls second.

    THIRD ROLL
    Now we're left with 22 stats to choose from (23 minus CC), in addition to the 6 double stat rolls.
    That gives us a total of 28 events each with probability 1/28. Let's say +Critical Hit Damage rolls as the third property.

    FOURTH ROLL
    21 stats left, plus the 6 pairs. Total of 27 events each with prob 1/27. Now let's say the +AR roll occurs.

    FIFTH ROLL
    20 stats left, plus the 6 pairs. Total of 26 events each with prob 1/26. That's when the double stat Int & Vit roll happens. Now this has a total probability of Y = (1/30)(1/29)(1/28)(1/27)(1/26) = 1/570,024 of occuring.

    However, AS could have rolled last, CC first and AR third, etc. There are actually X = 5! = 5*4*3*2*1 = 120 possible ways to get the perfect trifecta through these 5 rolls. Since each has a probability of Y to occur, we end up with a probability of XY = 1/142506 to get the perfect gloves.

    That's 1 in every 142,506 trials.

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    Fair enough. Now let's pretend that you don't care about Vitality but still want AR to maximize your EHP. Then all you want is a roll of the following type:
      +inherent main stat (Int)
      +random property 1
      +AR
      +Trifecta


    The calculations are similar: Y = (1/30)*(1/29)*(1/28)*(1/27)
    However, instead of multiplying by 5!, here we must take into acount the number of choices for random property 1:
    For example, before anything has rolled, we have 5 ways to get say AS: either on the first roll, or on the second, or the third etc.
    Then 4 ways left to get CC. Then 3 ways for CD and finally 2 ways left to get AR. But for the last roll you can have anything besides AS, CC, CD, AR and +Int. That gives you 26 possibilities. So X = 5*4*3*2*26. In the end that gives you XY = 26/5481.

    That's about 1 in every 210 trials. Not too bad.

    What does that really mean? -->
  • http://i1306.photobucket.com/albums/s565/drag0re/TRIAR_zps09a5180e.jpg
  • Horizontal axis displays the number of trials. Vertical axis shows the probability that you will get at least one success after that many trials. So after about 210 trials, you have a 60% chance of success. From the graph we see that after 800 trials you're pretty much guaranteed to get your trifecta+AR.

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    Now what about the probability to get just the trifecta, regardless of the last two rolls? #softcoreproblems

    The math actually gets a little more complicated. A more refined calculation would actually take into account the fact that if say a double stat roll of the form +Int & +<other mainstat> occurs, then the second random property cannot be any of the remaing pairs of stat with Int, which cuts two branches from our probability tree. Now whether AS rolls before or after CD, and likewise for CC does not matter. So we really have to consider only (5 Choose 2) = 10 combinations. In other words, it's like having to pick five pebbles among a pool of 30 pebbles, three of which are black and the remaining ones are white, and asking in how many ways you can arrange them. Also note that we may now have double rolls occuring twice.

    So, let's look at all the cases separately and add their respective probabilities.

    EDIT: Turns out it takes an entire page of math, so HERE IS THE RESULT.

    On average you will get a trifecta once in every 110 trials.

    Here's the graph:
    http://s1306.photobucket.com/user/drag0re/media/trifectaalone_zps8ea94311.jpg.html

    Enjoy.
    Edited by dragore#1339 on 3/25/2013 10:19 PM PDT
    Reply Quote
    item rolls are a joke in this game

    ever expect to find a perfect echoeing fury with lifesteal and crit damage? on average you would need to find 3 trillion echoeing fury's.... that sounds (not) fun
    Reply Quote
    Ive crafted probably about 30 gloves and 30 ammys, never gotten a trifecta.
    Reply Quote
    03/19/2013 03:19 PMPosted by Ozan
    Ive crafted probably about 30 gloves and 30 ammys, never gotten a trifecta.


    I roll over 300 ammys and 100 gloves, and none was close to a trifecta ( my golds goes to void )
    Reply Quote
    WOW, nice man! I've been trying to do this analysis as well.

    I'm curious now what the odds are to craft the godliest amulet possible

    +Max average damage (Not sure on the range, I think it's 35-100, although that range might require multiple affixes, say 25-60 average and 40 max damage)
    +100+ Mainstat with vit
    +9% attack speed
    +100% crit damage
    +10% crit chance.

    Unfortunately, now we're out of affixes. I can't live without all res on my amulet.
    Edited by DarkCecil13#1982 on 3/19/2013 4:48 PM PDT
    Reply Quote
    attack speed is definately much rarer affix on amulets atleast 100s of crafts only a few have even rolled it
    Reply Quote
    I'm convinced there's a lower chance for cc / cd / as to roll on items. How many have I found farming? 3. How many items with arcane resist, all resist, armor have I found (that also have the possibility for trifecta stats)? Hundreds if not thousands. RNG my ***.
    Edited by DeadDragon#1455 on 3/19/2013 4:53 PM PDT
    Reply Quote
    03/19/2013 04:47 PMPosted by Abacus
    attack speed is definately much rarer affix on amulets atleast 100s of crafts only a few have even rolled it


    I'm sure it's just RNG playing tricks on your mind. 100 amulet crafts is not enough to figure out the probability of certain rolls. Ivé crafted probably 200 amulets and only one had attack speed. Personally, I'd rather have a perfect average damage roll. It's the same dps gain as 9% attack speed. Bigger crits = more efficient for me.
    Reply Quote
    did i get lucky on my ammy, only crafted 15 got that one on the 5th try lol, its almost trifecta but w/e
    Reply Quote
    03/19/2013 04:52 PMPosted by DeadDragon
    I'm convinced there's a lower chance for cc / cd / as to roll on items. How many have I found farming? 3. How many items with arcane resist, all resist, armor have I found (that also have the possibility for trifecta stats)? Hundreds if not thousands. RNG my ***.


    im right with you, ive found 1 trifecta rofl, and it was a pos very low stat rolls, no vit or main stat lol
    Reply Quote
    Took me about 150 tries to roll a trifecta on my gloves, with a double dex roll. I've rolled about 200 ammys tho and no luck yet.
    Reply Quote
    03/19/2013 04:52 PMPosted by DarkCecil13
    attack speed is definately much rarer affix on amulets atleast 100s of crafts only a few have even rolled it


    I'm sure it's just RNG playing tricks on your mind. 100 amulet crafts is not enough to figure out the probability of certain rolls. Ivé crafted probably 200 amulets and only one had attack speed. Personally, I'd rather have a perfect average damage roll. It's the same dps gain as 9% attack speed. Bigger crits = more efficient for me.

    no i have gotten 3-4 in not a 100 crafts 100s somewhere between 200 and 400 crafts i know because atk speed is what i look for befor any other stat as i want a hp/atkspeed/loh for follower as well as trifecta for myself with ihgher crit %
    Reply Quote
    WOW, nice man! I've been trying to do this analysis as well.

    I'm curious now what the odds are to craft the godliest amulet possible

    +Max average damage (Not sure on the range, I think it's 35-100, although that range might require multiple affixes, say 25-60 average and 40 max damage)
    +100+ Mainstat with vit
    +9% attack speed
    +100% crit damage
    +10% crit chance.

    Unfortunately, now we're out of affixes. I can't live without all res on my amulet.


    The calculations would go along the same line of the first case for the gloves, and would probably be about the same order if we ignore the precise values that are rolled for each stat that you have listed here. Now if you're asking about the probability of getting, on top of these 5 stats, the best possible numbers, then we're probably talking about more than several millions of trials - not worth counting on.
    Edited by dragore#1339 on 3/19/2013 6:28 PM PDT
    Reply Quote
    Please tell me if I am wrong, but if you wanted four specific stats (tri+ar+random) wouldn’t the probability be (4choose4)(26choose1) / (30choose5) = 26/142506 or about 1 in 5481.

    And the case where you just want the trifecta be (3choose3)(27choose2) / (30choose5) = 351/142506 or about 1 in 406.
    Reply Quote
    Please tell me if I am wrong, but if you wanted four specific stats (tri+ar+random) wouldn’t the probability be (4choose4)(26choose1) / (30choose5) = 26/142506 or about 1 in 5481.

    And the case where you just want the trifecta be (3choose3)(27choose2) / (30choose5) = 351/142506 or about 1 in 406.


    I believe this is incorrect for the following reason: here you are counting the number of combinations, "(X choose Y)", but combinations do not care about the ordering of the chosen objects. For example, the number of ways to combine 2 out of the 3 letters A B C is (3 choose 2) = 3.

    Those combinations are AB, AC and BC.

    But in our case, our probability tree must also consider all the possible permutations. So that AB is different than BA. Sure, it gives the same stats in the end, but there are now two possible ways to get em, which yields a slightly higher probability.
    Edited by dragore#1339 on 3/19/2013 6:27 PM PDT
    Reply Quote
    Posts: 1,061
    Hey OP does the calculation take into account the value of the affixes? My head explodes when looking at maths so had to ask :p
    For example 6/6/50 and 10/10/100 are both trifeca.
    Reply Quote
    I believe this is incorrect for the following reason: here you are counting the number of combinations, "(X choose Y)", but combinations do not care about the ordering of the chosen objects. For example, the number of ways to combine 2 out of the 3 letters A B C is (3 choose 2) = 3.

    Those combinations are AB, AC and BC.

    But in our case, our probability tree must also consider all the possible permutations. So that AB is different than BA. Sure, it gives the same stats in the end, but there are now two possible ways to get em, which yields a slightly higher probability.


    True, I am saying that AB and BA are the same. And I believe I can do so as long as I do the same for the denominator. I am simply making no distinction between order throughout.

    Okay, so for your Trifecta+AR+1 random case, you can account for order like you did and say there are 5*4*3*2*26 = 3120 ways to make a trifecta but the total number of ways to make a 5 prop item is 30*29*28*27*26 = 17100720 when order matters. Therefore P(tri+ar+random) = 3120 / 17100720 = or about 1 in 5481 (which is what I got use combinations).

    *Edit formatting
    Edited by PeeMcGee#1334 on 3/19/2013 7:50 PM PDT
    Reply Quote

    Okay, so for your Trifecta+AR+1 random case, you can account for order like you did and say there are 5*4*3*2*26 = 3120 ways to make a trifecta but the total number of ways to make a 5 prop item is 30*29*28*27*26 = 17100720 when order matters. Therefore P(tri+ar+random) = 3120 / 17100720 = or about 1 in 5481 (which is what I got use combinations).

    *Edit formatting


    This is almost correct. Your numerator, aka the number of ways to get that one particular event (AR+Trifecta) is indeed equal to 3120.

    Your denominator, however, which is equal to the total size of the sample space (or as you put it, the total # of ways to make a 5 prop items) is not (30 Permute 5).

    It would be if some of these properties were not mutually exclusive. However, consider what happens if the first propery rolls +Strength, and the second one rolls +Str & + Dex. Then for the third property, we do not have 28 choices left, but 26. That is because +Str & +Vit and +Str & +Int may no longer roll, since +Str has already rolled twice (first from a single stat, second from a double stat roll). Same idea for Dex and Vit, which removes several branches from our probability tree - thus making the denominator smaller than what you got.
    Edited by dragore#1339 on 3/19/2013 9:09 PM PDT
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