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Unchained loot: updated drop statistics

September 25, 2016 1 comment

Here is an updated version of my previous tables about loot that drops from the various kinds of caches that you get from unchained and excavator dungeons. As a bonus, I have now included a table for plain old Excavator’s Kits for the first time!

Excavator’s Kit (green)

The other caches in this post were introduced with the unchained dungeon system in Update 4.1.4, but Excavator’s Kits are older, having been introduced together with House of Crom in Update 3.2. They drop directly from bosses and even trashmobs in unchained and excavator dungeons, and judging by what we’ve seen on testlive, they will also drop in the new Slithering Chaos dungeon.

The table below is based on 1084 drops from 719 Excavator’s Kits (I got one drop 385 times; two drops 303 times; three drops 31 times). There doesn’t seem to have been any change in the number of drops per kit in Update 5.0.

What How many times it dropped
Green potions: 235
3 Daggamalt 83
3 Sweetpressed Haste 76
3 Moonspill 76
Blue potions: 71
2 Potent Daggamalt 22
2 Dire Sweetpressed Haste 24
2 Absolute Moonspill 25
Food: 204
Ta Neheh Leaf Elixir 70
Kingsmight Ale 69
Bloodpurple Ale 65
Other:
Small money drop (48–88 copper) 238
Large money drop (3.9–15.5 silver) 10
an old-world level 80 blue BoE item (not from one of the level 70–80 sets)

46
an old-world level 80 purple BoE item (I got a Blade of Black Bile)

1
Manual of Discipline [10000 Mastery AA XP]

214
Flask of Completion [20000 Mastery AA XP]

65

Each type of money drops seems to be roughly uniformly distributed in its range. For the small drops, the minimum I got was 0.4813 silver, the maximum was 0.8830 and the average was 0.6778. For the large drops, the minimum was 3.9099, the maximum was 15.5092 and the average was 9.7149.

Unopened Chest (green)

What How many times it dropped
3 Daggamalt

119
3 Sweetpressed Haste

114
3 Moonspill

106
Manual of Instruction [2000 Mastery AA XP]

126
Manual of Discipline [10000 Mastery AA XP]

69
Flask of Completion [20000 Mastery AA XP]

34
2 Ta Neheh Leaf Elixir

21
2 Kingsmight Ale

17
2 Bloodpurple Ale

16
money (1–5 silver)

285
an old-world level 80 blue BoE item (not from one of the level 70–80 sets)

15
an old-world level 80 purple BoE item (I got a Black Cragsfall)

1
Pet (9× Shredder, 7× Acheronian Raider, 5× Undead Guardian, 4× Emperor Scorpion)

25
Pharaoh’s Guard green armor (1× Armplates, 1× Boots, 2× Gloves, 1× Goldmail, 1× Helm, 3× Sleevelets)

9

When we say that several of these things can drop, they can be the same thing twice, so you might get e.g. 6 Daggamalt from one chest. if the game decides to give you two money drops from the same cache, they will be shown separately in the money channel, rather than added up and shown as a single drop.

The table above shows 957 drops from opening 728 caches. It appears that the number of drops per cache has been increased in Update 5.0 (this also applies to Acheronian Caches and Mystical Acheronian Caches), at least for subscribers. Before 5.0, I got from 1 to 3 drops per cache (1 drop from 535 caches; 2 drops from 86 caches; 3 drops from 2 caches); since 5.0, I got from 2 to 4 drops per cache (2 drops from 76 caches; 3 drops from 24 caches; 4 drops from 5 caches).

It seems that potion and food drops don’t take your class into account, so that I get mana potions and food on my guardian as well. (Initially I thought that food drops take your class into account, but eventually I started getting mana food on my guardian as well; perhaps it was just a matter of randomness or they may have changed that in 4.1.5 or thereabouts.)

From the money drops seen so far, it seems pretty reasonable to conclude that the amount of money in each money drop is distributed uniformly in the range [1, 5] silver. (The actual minimum and maximum I’ve seen were 1.0062 and 4.9796 silver.)

Acheronian Cache (blue)

Here are the results based on 361 drops from 260 caches. Like with the green caches above, the number of drops per cache seems to have been increased in Update 5.0. Before that update, I got 1 or 2 drops per cache (1 drop 165 times; 2 drops 85 times); after that, I got from 2 to 4 drops per cache (2 drops 6 times; 3 drops twice; 4 drops twice).

What How many times it dropped
money (19–50 silver)

22
Food:

47
2 Ta Neheh Leaf Elixir

27
2 Kingsmight Ale

20
Blue potions:

6
2 Dire Sweetpressed Haste

6
Green armor:

37
Pharaoh’s Guard Armplates

6
Pharaoh’s Guard Boots

4
Pharaoh’s Guard Gloves

5
Pharaoh’s Guard Goldmail

6
Pharaoh’s Guard Helm

9
Pharaoh’s Guard Shenti

2
Pharaoh’s Guard Sleevelets

5
Blue armor:

185
Belt of Red Ruin

14
Belt of the Fallen Empire

21
Greaves of Red Ruin [feet]

23
Greaves of the Fallen Empire

23
Rerebrace of Red Ruin [shoulder]

25
Rerebrace of the Fallen Empire

23
Tasset of Red Ruin [legs]

29
Tasset of the Fallen Empire

27
Purple armor:

7
Gloves of Red Ruin [hands]

1
Vambrace of Red Ruin [wrist]

2
Vambraces of the Fallen Empire

4
Buffs:
(41× Elixir, 6× Refined Elixir, 2× self-rez)

49
Elixir of Brute Force [366 combat rtg]

7
Elixir of Guile [73 crit rtg]

8
Elixir of Invigorative Rejuvenation [15 nat stam regen]

13
Elixir of Precision [73 hit rtg]

13
Refined Elixir of Brute Force [732 combat rtg]

1
Refined Elixir of Guile [183 crit rtg]

1
Refined Elixir of Invigorative Rejuvenation [30 nat stam regen]

1
Refined Elixir of Precision [183 hit rtg]

3
Minor Elixir of Resurgence [self-rez]

2
Pets:

8
Mini-Pet: Acheronian Bloodletter

1
Mini-Pet: Acheronian Ritualist

2
Mini-Pet: Demigod

2
Pet: Living Statue

3

Regarding money drops, the ones I’ve seen so far lead me to believe that the amount is uniformly distributed in the range [19, 50] silver or so. The average of the drops I’ve seen so far is 34.19 silver. The only problem here is that my notes show drop drops far outside this range, namels 10.4923 and 10.8027, but that might well be an error in my notes. Apart from those two outliers, the lowest drop I’ve seen is 19.4109 and the highest is 49.3365.

I’m surprised that I got blue stamina potions (Dire Sweetpressed Haste) but no blue health potions (Potent Daggamalt). Mystical caches drop both, as we’ll see below.

The elixirs shown above are what you can get on a guardian; other classes get whatever is suitable for them (e.g. (Refined) Elixir of Mystical Excellence for mages).

Mystical Acheronian Cache (purple)

Here are the results based on 249 drops from 209 caches. Before Update 5.0, I was getting from 1 to 3 drops per cache (1 drop 172 times; 2 drops 30 times; 3 drops once); after that update, I got 2 or 3 drops per cache (2 drops 4 times; 3 drops twice).

What How many times it dropped
Blue potions:

14
5 Potent Daggamalt

8
5 Dire Sweetpressed Haste

6
Purple armor:

53
Breastplate of Red Ruin

2
Cuirass of the Fallen Empire [chest]

5
Gauntlets of the Fallen Empire

6
Gloves of Red Ruin

9
Helm of the Fallen Empire

5
Mask of Red Ruin

4
Vambrace of Red Ruin [wrist]

10
Vambraces of the Fallen Empire

12
Other purple bind-on-pickup gear:

12
Clasp of the Fallen Empire [necklace: 68 str, 37 hate inc rtg, 192 protection]

1
Emblem of Red Ruin [necklace: 86 str, 42 con, 18 hate dec rtg, 67 crit dmg rtg]

4
Trinket of Old Acheron [necklace: 534 combat rtg, 50 fatality rtg, 28 hate dec rtg, 67 crit dmg rtg]

3
Ring of Ancient Python [52 con, 52 hit rtg, 267 protection]

1
Ring of the Fallen Empire [80 str, 50 con, 35 hate inc rtg]

2
Claw of the Death Master [1hb: 116.1 dps, 86 str, 47 hit rtg, 47 crit rtg, 55 crit dmg rtg]

1
Purple bind-on-equip gear:

3
Fellblade of Crimson Slaughter [2he, boe: 132.4 dps, 42 str, 91 magic dmg (fire), 337 combat rtg (2he), 38 hit rtg, 72 crit rtg, 12 hate dec rtg, 90 crit dmg rtg, 18 mana tap rtg] 1
Fellhammer of the Sanguine Disciple [2hb, boe: 133.3 dps, 67 str, 45 con, 391 combat rtg, 38 hit rtg, 60 crit rtg, 90 crit dmg rtg] 1
Heater-Shield of the Fallen Empire [shield, boe: 890 armor, 36 str, 52 con, 24 hate inc rtg] 1
Buffs:
(38 Refined Elixir, 99 Refined Philtre, 18× self-rez)

155
Refined Elixir of Brute Force [732 combat rtg]

8
Refined Elixir of Guile [183 crit rtg]

9
Refined Elixir of Invigorative Rejuvenation [30 nat stam regen]

9
Refined Elixir of Precision [183 hit rtg]

12
Refined Elixir of Strength [183 hit rtg]

1
Refined Philtre of Constitution [+5% con]

47
Refined Philtre of Strength [+10% str]

52
Minor Elixir of Resurgence [self-rez buff]

18
Pets:

7
Pet: Scorpion Archer

3
Mini-Pet: Un Nefer

2
Mini-Pet: Scorpion Abomination

2
Other:

Phial of Tranquility [1 Expertise point]

5

One particularly annoying detail: not one, but TWO hate decrease necklaces (both bind-on-pickup, of course) can drop for a guardian. Perhaps they were inspired by John Donne‘s bitter but delightfully snarky lines:

  My constancy I to the planets give;
  My truth to them who at the court do live;
  My ingenuity and openness,
  To Jesuits; to buffoons my pensiveness;
  My silence to any, who abroad hath been;
      My money to a Capuchin:
    Thou, Love, taught’st me, by appointing me
  To love there, where no love received can be,
Only to give to such as have an incapacity.

On my ToS, I got an Amulet of the Hell Walker and Aegis of the Blood God; on my assassin, I got an Amulet of Ancient Python.

Mystical Excavator’s Kit (purple)

These are the purple caches that you get as a quest reward for the excavator quest (to clear Ardashir Fort, Vile Nativity, Sepulcher of the Wyrm, and the Coils of Ubah Kan). Here are the results after opening 99 purple caches on my guardian, for a total of 120 drops. All this was before Update 5.0; there were 1 or 2 drops per cache (1 drop 78 times; 2 drops 21 times). I haven’t opened any caches of this type after 5.0 yet, so I don’t know if the number of drops has been increased here as well.

What How many times it dropped
Blue potions:

8
5 Potent Daggamalt

6
5 Dire Sweetpressed Haste

2
Purple bind-on-pickup weapons:

8
Arakh of the Archaeologian [1he, bop: 117.6 dps, 83 str, 50 hit rtg, 34 crit rtg, 34 hate inc rtg, 47 crit dmg rtg] 4
Halberd of the Archaeologian [polearm, bop: 142.7 dps, 169 str, 221 combat rtg, 100 hit rtg, 86 crit rtg, 55 hate inc rtg, 123 crit dmg rtg] 4
Purple bind-on-equip weapons:

4
Fellhammer of the Sanguine Disciple [2hb, boe: 133.3 dps, 67 str, 45 con, 391 combat rtg, 38 hit rtg, 60 crit rtg, 90 crit dmg rtg] 1
Greatsword of the Death Master [2he, boe: 131.3 dps, 123 str, 169 combat rtg (2he), 35 hit rtg, 70 crti rtg, 100 crit dmg rtg] 1
Partizan of Red Ruin [polearm, boe: 135.8 dps, 50 str, 40 con, 7.7 nat stam regen, 409 combat rtg, 50 hit rtg, 75 crit rtg, 80 crit dmg rtg] 1
Scimitar of Red Ruin [1he, boe: 111.7 dps, 60 str, 84 combat rtg (1he), 30 hit rtg, 20 crit rtg, 40 crit dmg rtg, 21 offhand rtg] 1
Buffs:
(36× Refined Elixir, 43× Refined Philtre, 6× self-rez)

79
Refined Elixir of Brute Force [732 combat rtg]

13
Refined Elixir of Guile [183 crit rtg]

2
Refined Elixir of Invigorative Rejuvenation [30 nat stam regen]

10
Refined Elixir of Precision [183 hit rtg]

11
Refined Philtre of Constitution [+5% con]

17
Refined Philtre of Strength [+10% str]

26
Minor Elixir of Resurgence [self-rez buff]

6
Pets:

11
Companion: Atlantean Shade

2
Companion: Excavator

1
Mini-Pet: Deep Walker

1
Mini-Pet: Forsaken Child of Yig

3
Mini-Pet: Queen Cao-Polyphya

1
Mini-Pet: Serpent Man Underling

1
Pet: Atlantean Shade

1
Pet: War Rhino Calf

1
Other:

Phial of Tranquility [1 Expertise point]

4

The Halberd of the Archaeologian uses the same model as Polearm of the Black Pharoah (from Coils of Ubah Kan).

The Partizan of Red Ruin uses the same model as Blighted Halberd (from Xibaluku) and Imperial Acheronian Halberd (from the Iron Tower). A recolored version of the same model is also used for the Whispering Touch (T3 crafted polearm).

In terms of stats, the bind-on-pickup polearm (Halberd of the Archaeologian) is very nice; basically the same stats as on the T4 polearm, just in slightly lower amounts. The bind-on-equip polearm (Partizan of Red Ruin) is much less attractive; compared to other purple polearms in the game, it’s low on strength and constitution and it seems to be a bit more dps-oriented. I guess you could say it’s somewhere on the level of T1/T2 polearms. At least it’s better than the other bind-on-equip purple polearms (The Herald of Blight, Return of Time).

On my necro, this type of cache also dropped a Dagger of the Archaeologian (BoP).

For a complete list of the various purple weapons and accessories that are available from the Mystical caches (of both types), see AoC > TV.

Categories: Age of Conan, Drop Rates

Obsidian Basilisk drop statistics

February 15, 2014 4 comments

In my experience, farming the basilisk was much more fun than the yeti boss the month before. They reduced the respawn timer to about 30–60 min and instances tended not to fill up before the boss actually spawned. As a result, I was able to get more than twice as many kills (141 on the basilisk, 63 on the yeti boss). Much like with the Black Dragon drop statistics from last month, the loot table contains a huge number of things and some of them never dropped for me at all, so the table below is just meant to give us a rough idea of the relative frequency of different types of drops. It shows what I got on my guardian from the Hunter’s Trove of the Obsidian Basilisk (blue caches). In addition, during this week, I also got four Slayer’s Caches of the Obsidian Basilisk (purple caches) directly from the boss kills, but these aren’t shown in the table below; nor are any results shown from kills on other characters.

Count

Item
Social animations, particles etc.
12

Electric Boogie, Bottled Spirits, Sticky Fresh Blood, Void Essence, Cowardice Remedy
9

Fatal Concoction
Food, potions, buffs
0

Potent Daggamalt, Dire Sweetpressed Haste
6

Ta Neheh Leaf Elixir
1

Kingsmight Ale
7

Elixirs (3 Guile, 3 Invigorative Rejuvenation, 1 Physical Might)
1

Philtre (1 Constitution)
1

Minor Elixir of Resurgence (self-rez)
Bags
5

Traveller’s Merchant Contract
Social armor sets
11

Heaven’s Lake set
2

Shattered Colossus set
3

Slayer’s set
0

other social armor sets
Other social armor
6

Crown of Spring
2

Tigerskin cloaks
Raid gear
23

T1 armor
6

T1 weapons (1 sword, 2 shields, 3 polearms)
7

T2 armor
2

T2 weapons (2 shields)
AA urns
8

Flask of Completion (20000 Mastery XP)
6

Flask of Direction (50000 Mastery XP)
2

Coffer of Radiance (62500 Mastery XP)
Pets
5

animal pets (1 Black Cobra, 2 Field Mice, 1 Fire Salamander, 1 Wolverine Kit)
2

dancing pets (1 Aquilonian male, 1 Khitan female)
Purple caches
2

Slayer’s Cache of the Black Dragon (purple cache)

Seen on other characters: 1× Fallen Leviathan set, 2× Heaven’s Lake set, 1× female Stygian dancer pet, 1× Topaz Komodo pet, 3× raid gear.

If we compare this with the Black Dragon drop statistics from a month ago, we can see many similarities and parallels, but also a few differences.

  • The blue health and stamina potions (Potent Daggamalt, Dire Sweetpressed Haste) seem to have been removed from the loot table.
  • Health food (Ta Neheh Leaf Elixir) is still much more common than stamina food (Kingsmight Ale).
  • Among the social sets, the plan seems to be that the current “regional” set drops a lot more often than the “global” sets. The Heaven’s Lake set is just as common here as the Sanctum Atlantis set was on the dragon.
  • There seems to be a kind of regional vs. global distinction for the non-set social items as well. For example, the tigerskin cloaks have been dropping from all three portent bosses so far, but I didn’t see any Atlantean Spellbinder’s Tunics or Tarantia-style face masks from the basilisk. But on the other hand, I didn’t see any corresponding new Khitai-themed items either.
  • In the crappy novelty consumable department, there seems to be one item that predominates on each boss. On the dragon, that was Mitra’s Blessed Ointment; here on the basilisk, it seems to be the Fatal Concoction, which AFAIK hasn’t been dropping from the first two portent bosses. Several of the items that used to drop from the dragon (Ritual Knife, Prayer Book, Fermented Brain Juices) don’t seem to be dropping from the basilisk.

Purple cache statistics

This table shows, for each boss, the total number of kills I’ve done (on all characters, regardless of class), the total number of purple caches I got (either as drops from blue caches or directly from the boss kill), and the things that dropped out of those purple caches. The “Simple Relics” column shows the number of times that I got simple relics from purple caches, followed by the total number of simple relics thus obtained. So for example, from the Dragon’s purple caches, I got simple relics seven times and I got a total of 29 simple relics that way. I never got any Rare Relics but I heard that some other people did, so I reserved a column for them as well.

Boss Total
no. of
kills
# purple caches Drops from purple caches
from
blue
direct total Simple
Relics
Urns Rare
Relics
Pets
Dragon

185

6

2

8

7 (29)

0

0

1
Yeti

61

3

2

5

4 (15)

0

0

1
Basilisk

141

4

2

6

2 (7)

2

0

2

Money drops from mobs

September 12, 2012 5 comments

Many mobs drop some money in their loot bags. Usually these are pretty small amounts, but bosses in epic instances of playfields can drop a substantial amount. In fact, in the early months of the game, farming bosses in epic playfields was the major source of income for some players. If you killed a level 80 boss in epic Kheshatta with a group of 6 players, you could expect around 15 to 30 silver per player; and you would get a similar amount if you killed a level 50 boss in epic Field of the Dead alone. I even heard of level 40 players farming level 20-ish bosses in epic Wild Lands of Zelata for money with which they would buy their first horse.

A few months after release, Funcom nerfed the money drops from epic bosses by a factor of 5 or so — the same kills that would formerly net you 15–30 silver will now give you just 3–6 or so. Since then, farming epic bosses for money doesn’t make much sense. However, recently, as I was farming epic Kheshatta bosses for the sake of vanity gear (screenshots of which will be published in a forthcoming post as soon as I manage to transfer a few characters to testlive so I can take the screenshots there), I noticed that if you kill these bosses alone or teamed up with just one other player, the amount of money you get is still pretty decent. I probably made more than 50 gold while farming for my 12 epic Kheshatta armor sets.

So I became interested in the amount of money dropped by the bosses. Obviously this is a random number; but what can we say about its distribution? I started writing down the amount of money for each boss kill and ended up with the following statistics (all money amounts are in silver):

Boss
level
Number
of kills
Average
drop
Std. dev. Minimum Maximum
75 8 22.38 7.77 11.09 30.53
80 96 26.45 7.73 13.17 38.72
81 7 28.25 9.37 14.10 37.71
82 80 23.42 7.38 12.63 38.70

Now, obviously we don’t have enough data for level 75 and 81 to say anything reliable there. And even for level 80 and 82, where we have a decent amount of data, we see that surprisingly the average drop from level 82 bosses is smaller than from level 80 bosses — this is surely a sign that we still have insufficient data to estimate the averages reliably enough. Still, we have enough data to plot a rough histogram. We divided the range from min to max into 10 smaller ranges (of equal width); the histogram shows, for each subrange, how many kills gave us an amount of money that falls into that particular subrange:

I was expecting to see something bell-shaped, like a normal distribution — but we see that this is not the case. The values around the average don’t really seem any more likely than those closer to the extremes. And the extremes themselves, the maximum and minimum, are suspiciously close to a ratio of 3:1 — a nice round number which we would hardly dare to expect from e.g. a normal distribution, but which wouldn’t be surprising if it’s a uniform distribution and the developers deliberately chose its range so that max = 3 × min.

Farming level 50 bosses

Now, it’s a pain in the ass to get more data for the above table, since killing level 80 epic bosses takes a lot of time; and I don’t intend to kill any more of them now that I’m done collecting my epic Kheshatta armor sets. So I started killing level 50 bosses in epic Field of the Dead instead, hoping that the distribution of money drops there is the same (just with smaller values, of course). Here are the results after 500 boss kills:

Boss
level
Number
of kills
Average
drop
Std. dev. Minimum Maximum
50 500 5.14 1.48 2.56 7.67

These are just the sort of things that you would expect from a uniform distribution. A uniform distribution on the range [a, b] will have the average μ = (a + b)/2 and the standard deviation σ = (ba) / sqrt(12). So if we take a = min = 2.56, b = max = 7.68, we see that the average should be 5.12 and the standard deviation should be 1.478, which is very close to what we see in the table above. And we can also see that the maximum is roughly three times the minimum, b = approx. 3a, similar to what we already saw above in the case of level 80 bosses.

The histogram also shows how close we are to a uniform distribution:

Of course there’s some variation from column to column, but that’s only to be expected.

So, at this point we can speculate that the amount of money dropped by an epic boss of a given level is distributed uniformly in the range [a, 3a], where a depends on the level. It would be interesting to kill a few bosses of other levels to get a better idea of how a depends on the level (my guess is that it’s an exponential function of the level), but that would be pretty time-consuming as it’s hard to find a lot of bosses at the same level. Level 50 is a bit of an exception there, because you have the Fields of the Dead with 13 level 50 bosses in a small area; for most other levels you’re lucky if you can find one or two bosses at that level.

Incidentally, I managed to kill 120 level 50 bosses in epic Fields of the Dead in 90 minutes of farming (I think with a bit of care you could do it faster still). At an average of 5.14 silver per kill, this gives you an income of 4.11 gold per hour — not bad at all compared to things like doing quests in Kara Korum or farming resources for sale (at current prices on the Crom server at least). But it does assume that you’re the only player farming there; and I saw other people killing those bosses more often than I had expected.

By the way, while I was killing the level 50 bosses, I also wrote down which blue items dropped that weren’t part of any of the level 40–69 world-drop sets. I ended up noticing the following 20 items (they’re all bind-on-equip):

  • Battlebrawn Necklet
  • Battlespite
  • Backbreaker
  • Blacksever
  • Bladebrave Tunic
  • Bloodblight Bolts
  • Bloodrighteous Belt
  • Bloodpurge Boots
  • Compendium of Many Hurts
  • Corpseskewer
  • Howler Hide Cloak
  • Mark of Atrocity
  • Mark of Hate
  • Mindward Robe
  • Painthreaded Leggings
  • Sightshift Mantle
  • Sleekspeed Leggings
  • Soulfeast
  • Sparkfrost
  • Vigorspine Tunic

The last time I added a new (previously unseen) item was after approx. 380 boss kills. So it’s entirely possible that there might be a few more such items dropping there that I haven’t seen even after 500 boss kills.

Anyway, the number of such out-of-set items in the bosses’ loot table is potentially interesting if you want to estimate how long you’ll have to farm them for. In my post about the level 40–69 sets most of the calculations simply pretended that the loot table contains just the 12 × 8 = 96 items from the sets and nothing else; now we see that a more reasonable size of the loot table would be around 116, not just 96.

More about the uniform distribution

Suppose that we have a sample of n points, X1, …, Xn, taken independently from a uniform distribution in the range [a, b]. The mean of such a distribution is μ = (a + b)/ 2 and its variance is σ2 = d2 / 12, where d = ba. (You can easily derive these formulas for μ and σ2 by yourself from the definitions of mean and variance, or you can look them up in the Wikipedia.) In practice we don’t really know any of a, b, μ and σ — we just know the sampled values X1, …, Xn, and the interesting question is how to find out something about a, b, μ and σ from our sample.

We can start by computing things like:

  • Sample average: Xa = (1/n) Σ1 ≤ in Xi.
  • Sample variance: S2 = (1/(n − 1)) Σ1 ≤ in (XiXa)2.
  • Sample maximum: Y = max {X1, …, Xn}.
  • Sample minimum: W = min {X1, …, Xn}.

Since our X1, …, Xn are random variables, the above-listed sample statistics Xa, S2, Y and W are random variables as well. What can we say about their distributions?

As is well known, the sample average is distributed approximately according to the normal distribution with the expected value μ and variance σ2 / n. Since its expected value is μ, this means that the sample average is an unbiased estimator of μ. The standard deviation (square root of the variance) tells us that on average, the sample average will deviate from the correct value of μ by √(σ2 / n) = σ / √n.

How much is that in practice? In the case of our 500 level 50 boss kills, this standard deviation amounts to approx. 6.6 copper; for the 96 level 80 boss kills, the standard deviation was approx. 79 copper.

If we move on to the sample variance, S2, it is similarly well known that its expected value is σ2; in other words, it is an unbiased estimator of the population variance. (That’s why we had to divide by n − 1 when computing S2; if we had divided by n instead, which might seem to be intuitively more reasonable at first sight, we would have obtained a biased estimator that would slightly overestimate the population variance.)

Next let’s take a look at the sample maximum, Y. We can see that its cumulative probability function will be

P(Y < y) = P(X1 < y, …, Xn < y)
= P(X1 < y, …, Xn < y)
= P(X1 < y) · … · P(Xn < y)
= [(ya) / d]n.

We now have to take its derivative with respect to y to obtain the probability density function:

pY(y) = n (ya)n − 1 / dn.

Now we’re in a good position to calculate the expected value of Y:

E[Y] = ∫ab y pY(y) dy
= … = bd / (n + 1).

You can see right away that this is a little less than b, the true maximum of our distribution; but the discrepancy gets smaller if we take a bigger sample (i.e. a bigger n). And intuitively it certainly makes sense that the sample maximum, Y, underestimates the population maximum b, because sometimes Y is below b (if all your samples Xi are below b), but this doesn’t get counterbalanced by cases when Y would be above b (because this is impossible — none of the Xi can be greater than b, so Y can’t be either). So we see that using the sample maximum Y, to estimate the population maximum b will actually cause us to underestimate it. Soon we’ll see how to get a better estimator.

Finally we have the sample minimum, W. Here we can use a very analogous way of thinking as we did for Y above and we would end up with the following result:

E[W] = a + d / (n + 1).

In other words, on average, W overestimates the population minimum a by just as much as Y underestimates the population maximum b.

Estimating a and b from W and Y

Now, as is often the case in statistics, we can try to derive estimators of a and b by pretending that our Y and W happened to achieve exactly their expected value and then solving the resulting equations for a and b. In other words, we have the pair of equations

Y = bd / (n + 1)
Z = a + d / (n + 1).

Remembering that d = ba and solving these equations for a and b gives us the following estimators:

A = W − (YW) / (n − 1)
B = Y + (YW) / (n − 1).

Again, intuitively this makes a lot of sense; we know that Y underestimates b, so to get a better estimate of b we must add something to Y (and similarly we must subtract something from W). We know that Y was a biased estimator for b because on average it underestimated it; but what about B, is it still biased or do we finally have an unbiased estimator now? To find out, we have to compute its expected value:

E[B] = E[Y + (YW) / (n − 1)]
= E[Y n / (n + 1) − W / (n − 1)]
= E[Y] n / (n + 1) − E[W] / (n − 1)
= … = b.

So we see that B is an unbiased estimator of b, and we could similarly show that A is an unbiased estimator of a.

Estimating μ from W and Y

Now, remember that the mean of a uniform distribution is μ = (a + b) / 2. Since A is an unbiased estimator of a, and B is an unbiased estimator of b, it follows that (A + B) / 2 is an unbiased estimator of (a + b) / 2, i.e. of μ. Let’s call it M:

M = (A + B) / 2 = (W + Y) / 2.

This is the second unbiased estimator of μ we’ve seen today — the first one was the plain old sample average, Xa. Which of these two estimators is better? The fact that they are unbiased really just means that their mistakes in both directions tend to balance each other out on average (i.e. they err just as much by overestimating as they do by underestimating); this doesn’t yet tell us anything about how big these errors are, on average. For that we need to compute their variance. We already saw earlier that the variance of Xa is D[Xa] = σ2 / n = d2 / (12 n). But what about M?

The variance of M is a bit more tricky to compute. We saw that M can be computed as W + Y, but these two variables are obviously not independent of each other (e.g. because we know that WY), so we can’t say that D[M] = D[W] + D[Y]. Instead, we can start by computing the joint probability distribution of W and Y. The cumulative probability function is

P(Y < y, W < w)
= P(Y < y) − P(Y < y, Ww)
= P(Xi < y for all i) − P(wXi < y for all i)
= [(ya) / d]n − [(yw) / d]n.

To obtain the probability density function, we need to take the partial derivative with respect to y and w:

p(y, w) = ∂2 P(Y < y, W < w) / (∂yw)
= … = n (n − 1) (yw)n − 2 / dn.

(All of this of course only makes sense when w < y. If w is greater than y, the probability P(Y < y, Ww) is 0, as the sample minimum cannot be greater than the sample maximum. In that case P(Y < y, W < w) = P(Y < y) and since it no longer depends on w, its partial derivative with respect to w will be 0, so that p(y, w) will also be 0 there.)

Now we’re in a good position to start computing the variance of M:

D[M] = E[(ME[M])2]
= E[((W + Y) / 2 − (a + b) / 2)2]
= (1/4) E[(W + Yab)2]
= (1/4) ∫∫awyb (w + yab)2 p(y, w) dy dw
= n (n − 1) / (4 dn) ∫∫awyb (w + yab)2 (yw)n − 2 dy dw.

This looks a bit hairy, but it becomes a lot easier when we substitute the variables a bit. Let’s introduce u = yw and v = w + yab. Our integral turns into

D[M] = n (n − 1) / (4 dn) ∫0d duuddu dv (1/2) v2 un − 2
= … = d2 / [2 (n + 1) (n + 2)].

Remember that our other estimator of μ, namely Xa, had a variance of D[Xa] = d2 / (12 n). So we can see right away that M has a smaller variance by approximately a factor of n / 6. Or in other words, if we use M instead of Xa to estimate μ, our errors will be on average √(n / 6)-times smaller. Or in still other words, if we want to make sure that our average error will be sufficiently small, we need a much smaller sample size (i.e. n) if we use M than if we use Xa.

Estimating a and b from Xa and S2

We saw earlier that for a uniform distribution, the mean is μ = (a + b) / 2 and the variance is σ2 = (ba)2 / 12. If we solve this pair of equations for a and b, we see that a = μ − σ √3 and b = μ + σ √3. Since we know that e.g. Xa is an unbiased estimator for μ, and S2 is an unbiased estimator for σ2, we can plug them into these formulas for a and b to get another pair of estimators for a and b:

A’ = XaS √3
B’ = Xa + S √3.

Are the new estimators unbiased? Let’s try:

E[B’] = E[Xa + S √3]
= E[Xa] + √3 E[S]
= μ + √3 σ = b,

and similarly we could see that E[A’] = a. So these new estimators are also unbiased, just like our previous ones (A and B) were.

An interesting question would be which pair has the smaller variance; but when I tried to compute the variance of A and B, I couldn’t get any elegant results, it all quickly turned into a horrible mess.

However, there is one other argument against using A’ and B’: it’s perfectly possible that A’ might turn out to be greater than W, and B’ can be less than Y. And if that happens, you’ll feel a bit silly having estimated the population maximum with a number that is smaller than your sample maximum. This cannot happen with A and B.

Wrapping up

Now that we know a bit more about uniform distributions, we can go back to our experimental data:

Statistic Boss level
50 75 80 81 82
n (sample size) 500 8 96 7 80
W (sample min) 2.56 11.09 13.17 14.10 12.63
Y (sample max) 7.67 30.53 38.72 37.71 38.70
Xa (sample avg.; estimator of population avg.) 5.14 22.38 26.45 28.25 23.42
D[Xa]* 0.066 2.74 0.79 3.54 0.82
M (better estimator of population avg.) 5.11 20.81 25.95 25.91 25.67
D[M]* 0.0072 1.86 0.19 2.62 0.23
A (estimated population min) 2.55 8.31 12.90 10.17 12.30
B (estimated population max) 7.68 33.31 38.99 41.65 39.03

*Note: actually, the formulas for D[Xa] and D[M] require d, which is ba; we of course don’t know the true values of any of these parameters, so we used BA instead.

The standard deviations √D of Xa and M are useful as they give us an idea of how (un)reliable our estimates of μ are, and also how much more reliable M is than Xa. You can see that the expected error of our M-estimate of the average drop from level 50 bosses is barely 72 tin!

Enigmata of Yag bosses

December 26, 2011 Leave a comment

The Enigmata of Yag is a popular group dungeon in Kara Korum. You get one of 3 possible bosses in it: Tetharos the Slaughterer, X’cth the Annihilator, and Spawn of Nyarlathotep. Tetharos is harder than the other two, whereas Spawn and Annihilator are both pretty easy but Spawn drops more loot and AA xp. Thus people often use rangers with the tracking ability to see which boss is in the instance before they start actually clearing it; if it isn’t the boss they want, they regroup and keep trying with new instances of the dungeon.

I’ve also had a ranger available for tracking for some time now, and I’ve been keeping count of how many times I saw which boss. The following table shows the results:

Boss

Count

Relative frequency

Tetharos the Slaughterer

309

47.4%

X’cth the Annihilator

300

46.0%

Spawn of Nyarlathotep

43

6.6%

(Total)

652

100.0%

So on average you’ll have to track about 15 times to find the Spawn! Keep that in mind next time you feel that you’ve been tracking unfairly long.

Shards of the Exiled God

December 24, 2011 6 comments

The Shard of the Exiled God is a purple tradeskill resource that is used to craft bags (Soldier’s Haversack, Commander’s Haversack) and epic weapons (Quill/Feather of Ibis in T2, and the various T3 crafted weapons). It drops from raid mobs in BRC Wing 1, between Yaremka and Sabazios.

  • There are 28 mobs that can drop shards. Non-raid mobs (i.e. the ones without skulls around their name), minions etc. don’t drop shards. Snakes from Sabazios’ room, and the Vengeance of Set from the bowl upstairs, AFAIK don’t drop shards (I’ve never seen it happen at any rate).
  • In addition to shards, these mobs can also drop armor pieces. It’s even possible for the same loot box to contain a shard and a piece of armor.

The statistics so far:

  • 451 shards dropped in 418 runs (in each run, all 28 mobs are killed).
  • That’s 1.079 shards per run.
  • For each mob, the probability that it will drop a shard is thus 1.079 / 28 = 3.85%. Let’s call this probability p.

If we assume, as is reasonable, that the loot table of each mob is independent of the other mobs, we see that the probability that k shards will drop in a run is binom(28, k) pk (1 – p)28 – k. We can compare these theoretical probabilities with the actually observed ones and see that they match very nicely, so our underlying model seems to be sound:

k (number of
shards)

Number of runs (out of 418) in which exactly k shards dropped

Observed probability of k shards

Theoretical probability of k shards
0 136 32.5 % 34.0 %
1 161 38.5 % 37.4 %
2 80 19.1 % 19.9 %
3 34 8.1 % 6.8 %
4 7 1.7 % 1.7 %
0 0 0.0 % 0.3 %

Some observations:

  • About a third of all runs are dry, i.e. without shards. The probability of three dry runs in a row is about 3.4% — if you farm long enough, it will happen. Keep that in mind when it happens, it feels horrible but doesn’t mean that anything is wrong with the loot tables or the random number generator 🙂
  • A greedy PUG raid leader that keeps the first shard to himself, ends up keeping 62% of all shards that drop. If he claims the first and third shard for himself, he ends up keeping 72% of all shards. If he claims the second shard for himself (a few decent ones do that), he ends up keeping 27% of all shards.

Some people say that Black Ring Scions have a higher probability of dropping a shard than other mobs do. (Indeed in the early days after the shards were introduced (in update 1.05), there was an impression that only the scions drop them.) I’ve never tried counting the drops from Scions separately from those of other mobs, so I don’t have any numbers to either support or disprove this claim. People who farm shards in ultra-small groups (like just 2 people) often kill just scions in the first room before resetting the instance.