30 January 2018

Diablo III: Primal Ancient Hellfire Chance

Correction Statement
So apparently the drop rate of primal ancients isn't 10% of the ancient rate as I picked up back when it came out. Additionally, the attributes or stats that are rolled for items aren't evenly distributed. Both these facts render the used variables and parameters of the calculation to being false. Since the amount of work to correct everything is higher than I'd like I'll leave this post as a 'what if' kind of post. I might pick up the topic again later someday using the data that was posted on Reddit. Thanks to everyone. 

I've always been a fan of mage classes. In Diablo III, however, I play a demon hunter. It's weird but the demon hunter is a lot more fun in my opinion. Especially the multishot build with the unhallowed essence set. Though there is a really annoying thing.

The Hellfire Amulet

Hellfire amulets, oh sorry, The Hellfire Amulet is a special item that can only be crafted. Being able to craft it requires you to buy a recipe from the Squirt. It doesn't stop there though, obviously, we need the crafting materials as well. The materials for this craftsmanship drop from the uber bosses. To fight the uber bosses you have to get the infernal machines from the keywardens and use them in the heretic's abode in Tristam in Act I. So we have to defeat the keywardens in Act I to Act IV, kill the uber bosses and then we can basically craft the hellfire amulet.


Unfortunately forging this hellfire amulet may or may not be in vain. The unique property of this item is that it gives you one of your class's passive abilities. Usually, you're limited to four passive abilities but thanks to the amulet you can have five. This is also the reason why it's so interesting for certain classes. Well as you may notice you'll want an amulet that has rolled one of those five passive abilities you want to take with you.

So let's check out our chances to get what we want. When we craft a hellfire amulet we get a hellfire amulet with 100% certainty. So the first limiting factor will be passive. There are 19 different passive abilities, the amulet has only one of these and 5 are okay. So the chance to get one of these five we want is 5 / 19 = 0.2632, 26.32%. Well, that sounds... doable.

Once we have our hellfire amulet for a while over time we start to build on ancient pieces. At some point you also want your hellfire to be an ancient item. So let's see what the chance is for us to get our passive and an ancient hellfire amulet. The chance for a legendary item to roll ancient is 10%. So we want two chances to hit at the same time. We can use stochastic to calculate this further.

A = "Chance to get our passive" P(A) = 0.2632
B = "Chance for it to be ancient" P(B) = 0.1

We want both at the same time so:

 P(A and B) = P(A) * P(B) = 0.2632 * 0.1 = 0.0263 = 2.63%

Our chances just dropped drastically.
Unfortunately, this is not the end.. I'm sorry but.. there's still the chance for it to be primal ancient. Primal ancient items are the creme de la creme, they always roll top value on all stats. However, there are two limiting factors for getting these.
  1. We need to have done a greater rift 70 or higher solo
  2. They have a drop chance of 0.01 or 1%
Well if we go with the odds our chance change from 2.63% to 0.2632 * 0.01 = 0.0026 or 0.26%

Still Not Done!

Even if we get a primal ancient item, it may have stats we don't want. We can only switch one stat but if we roll badly twice it'll look bad. Luckily there isn't much to roll around. There are five attributes on it aside from the passive ability. Two of these attributes are fixed. It will always have a socket, which we need and it will always have the main stat, for demon hunters its dexterity, that we choose when crafting. This means we have two of five variables we don't need to worry about. Leftover is the three others from which two are primary and one is secondary. Which stat combination is the best, of course, depends on your current stats, most often you will see amulets rolled with critical hit chance and critical hit damage as primary stats though. The secondary effect is your personal preference, never the less I'd assume reduced damage from ranged attacks would most likely be the best. To determine the chance we need to know how many options we want and how many options are available. For our secondary stats, there are 15 different attributes and we want one of those. So the chance to get this is 1 / 15 = 0.0667, 6.67%. For our primary attributes, we have each 16 available and we want two, however, once we got one of these our chance changes. This means our first primary attribute roll has a 2 / 16 = 0.125, 12.5% chance to be successful and our second one has a 1 / 15 = 0.0667, 6.67% chance to be successful.

For our perfect roll we want all possibilities to happen at the same time. Multiplying them we get:

0.2632 * 0.01 * 0.0667 * 0.125 * 0.0667 = 1.4637 * 10^-6, 0.0000014637 or 0.00014637%.

No Worries! We Still Have Rerolls

We can reroll one of the stats. One and only one. This means our chances improve! If we assume one of our primary attributes rolls successful and one doesn't we have a 2 / 16 = 0.125, 12.5% chance to get the successful one and our 1 / 15 = 0.0667, 6.67% chance for the other fails yet our secondary attribute succeeds with a 1 / 15 = 0.0667, 6.67% chance. So we have a 2 / 16 * 1 / 15 = 0.0083, 0.83% chance for this to be the case. If both our primary attributes roll successfully we have a 2 / 16 * 1 / 15 = 0.0083, 0.83% chance for this to occur. Adding both calculations together we get 2 / 16 * 1 / 15 * 2 = 0.0167, 1.67% . Now since we want a primal ancient item with the correct passive we do the following calculation:

0.2632 * 0.01 * 2 / 16 * 1 / 15 * 2 = 4.3867 * 10^-5, 0.000043867 or 0.0043867%

One Day

Don't worry if you won't be able to farm it this season, someday you'll have this on your account, forever, next to your 10000 paragon levels.
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I'm a B.Sc. Games Engineer and I created this blog to share my ideas, theorycrafting, thoughts and whatever I'm working on or doing.