Here is the graph for the TRIFECTA + AR:
Here is the graph for the TRIFECTA ALONE:
Y axis: probability to craft a trifecta...
X axis: ... after that many attempts.
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
- 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.
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.
This one is guaranteed.
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.
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.
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.
21 stats left, plus the 6 pairs. Total of 27 events each with prob 1/27. Now let's say the +AR roll occurs.
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.
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
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? -->
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: