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Rarest Item in Diablo - GhastMaster - 07-11-2020
Most of the end game items are easily findable given a few years of solid casual playtime. The bulk of them have a roughly 1 in 1000-2000 games chance of dropping. One exception would be a King's Bastard Sword of Haste with a 1 in 6500 games chance of dropping. Given that Griswold can sell them to you, this is not terribly difficult to obtain. One item that has always stood out is the Emerald Long War Bow of the Heavens at a whopping 1 in 16,200 games! It is arguably not necessary as you can use an Obsidian bow and attain maxed resistance, but it is very nice to find.
When thinking about the rarest item in the game, people often go to the double cursed items. As an example, Vulnerable Full Plate Mail of Weakness has a 1 in 304,000(!) games chance of dropping and it is only findable in Hell difficulty. There are other base armors that can drop that combination of affixes with the same probability: Plate Mail, Field Plate, and Gothic Plate. While they each individually have a 1 in 304,000, we need to divide by the 4 possible base items to see our chance of finding any one of those. That brings us to a 1 in 76,000 games chance of stumbling across one. Other double cursed affix combinations exist on these armors to give us the same probability of dropping. Rusted/Frailty, Vulnerable/Dyslexia, Vulnerable/Atrophy, Vulnerable/Disease, and many more. If we keep dividing we eventually come to a point where having an item with a 1 in 304,000 chance of dropping is not that rare in and of itself. There are 44 possible combinations. Ergo 304,000 / 44 = 6,909. Not that rare in the grand scheme of things. Granted, not too many people spend their time farming dlvl 2 in Hell for Rusted Plate of the Fool. With Emerald/Heavens only(arguably) sought after in the LWB form, I consider this more impressive find.
There is one affix combination that blows these out of the water: Emerald/Slaying. It has a 1 in 2,840,000 games chance of dropping for any individual sword, axe, or club you want to find it on. There are in total 25 possible base items. That brings us to a 1 in 113,600 chance of finding any combination of Emerald/Slaying in a game. That is greater than 16 times more rare than the double cursed armors!
This combination does not seem possible at first. Emerald has a qlvl of 31 and Slaying is 15. Items typically cannot have a prefix that is double the suffix qlvl or visa versa. There are exceptions with Wirt and Adria of course. There is also a well known quirk in Diablo coding that comes up quite often. Diablo rounds down. But whom could drop this you might ask? There is only one: Dreadjudge. With Dreadjudge his mlvl items is 27. You add 4 for bosses, bringing us to 31. So, he can drop items with a qlvl of 31. 31 is more than double 15, but 31 divided by 2 is 15.5. Diablo rounds this down to 15. He is the only boss that has an mlvl items of 27 and he only exists every 10 games. If he had an mlvl of 28, he would be able to drop items of qlvl 32 and below, but 32 divided by 2 is 16, so he would not drop Slaying.
Diablo uses a 32bit random number generator(system clock) for the game seed. The max possible game seeds there could be is 4,294,967,296. Assuming there are that many possible individual games, there should exist 2,526,451 individual Dragon's Amulet of the Zodiac. While as an example only 1,512 individual Emerald Bastard Sword of Slaying and 37,800 of all the Emerald/Slaying combinations exist!
These are the rarest items in the game. A close third is Emerald Shield of the Wolf, for similar reasons. I challenge you to decipher why it is third, knowing Wolf is the same qlvl as Slaying and there are only 6 types of shields.
RE: Rarest Item in Diablo - FireIceTalon - 07-13-2020
Pretty interesting stuff. Is a emerald sword of slaughter possible via Wirt, and if so up to what base. I know a basty would exceed the price cap.
RE: Rarest Item in Diablo - GhastMaster - 07-13-2020
My metric for determining the rarity was inconsistent. Also, I did not consider Wirt for whatever reason on some items. I think my brain was working too many ways at once. I may retract the whole post on my site and do a proper analysis. One mistake I made, of many, was not taking into consideration that each game seed can be used for normal, nightmare, and hell. Therefore, Drag/zod amulet actually has over 7,000,000 unique iterations. A person does not play three difficulties at once though!
Emerald Slaying bastard sword is possible at wirt with any stats, but not the great sword. Some of the emerald Shields of the wolf are also. I'm going to focus on dungeon only items and run some figures using 2^32 / norm rarity + 2^32 / night rarity + 2^32 / hell rarity for any single base item and affix combo that cannot be shopped. Items that cannot be found in shops and only on 1 difficulty should be "rarer". I might compare some of the affix combos taking into consideration how many base items they exist on.
Rarity can be defined two different ways. How much time relative to another item it would take to obtain and how many exist in all game seeds combined relative to one another. Time factor for players is clearly important, but the total number of iterations is interesting to me as well.
An interesting situation would arise if one item has a 1 in 6000 games chance of dropping in hell only and another has a 1 in 18,000 chance of dropping in each difficulty. They both have the same amount of unique iterations, but the latter is clearly more difficult to obtain.
RE: Rarest Item in Diablo - GhastMaster - 02-07-2021
Upon further review, Wirt cannot drop emerald slaying.