Bending the Curve - Adding a Long Tail to Your Random Rolls

The Logic of Chance, 3rd edition by John Venn

   The classic 2d6 and 3d6 die roll conventions provide a normal or Gaussian distribution, commonly referred to as a Bell Curve.  This is great when we are looking to simulate an evenly distributed range of scores, such as the classic six characteristics, but what about when we would like to randomly determine an NPC level?  That presents a use where the the bulk of the results should be towards the lower bound, while the upper bound is potentially indeterminate, in other words an asymmetrical curve.

     One of the classic asymmetrical curves is the Pareto Curve, which has been popularized as the 80-20 rule, where in management 80% of your problems are caused by 20% of your people; or 20% of your people do 80% of the work.  Helpdesk ticket closure times are another good example, where the bulk of the them are closed in a fairly short range of time, but others, especially those requiring equipment replacement, extend the overall time range far to the right.  Back when I was at GE Fleet Services, I would check the 3rd quantile, or the mark at which the technician had closed 75% of their tickets, rather than their average time to close.

     Looking at a 2d6 distribution we find, 97.22% probability that you roll less than a 12; 91.66% less than an 11;  83.53% less than a 10 and 72.42% less than a 9.  We can create a reasonable long tail distribution very simply.   Whenever an 11 or 12 is rolled, roll another d6 and add it to 10, producing a 2-16 distribution.

 

     3d6 distributions give me the opportunity to illustrate that you don't need to stay with the same size die when creating a long tail.   You have a 83.8% chance of rolling less than a 14 on 3d6.   So to create the long tail, I would roll another die whenever the 3d6 is 15 or more and add it to 14.   If I use a d6, it only gives a 3-20 distribution, not much of an increase over 3-18.  By using a d10 instead, I create a 3-24 distribution range, producing a significant tail at the high end.

     I used anydice to look at the probabilities.  Their calculator will allow you to see the percentages for any combination of dice.  

     I haven't come up with as simple a way to do a long left hand tail, but I don't have a use case either.  If I get one I'll probably just flip the scale at the bottom and read from right to left.

     As I said, the use case I identified is random NPC levels, I'd be interested in hearing of any other use cases you might come up with.

CSIO - Random Encounters Sailing the Roglaroon

    Adding to the area specific Random Encounter tables I developed last year, I've updated by Cross System Source Table to read.

d20	Game System		d20	Other System
01-10	AD&D		01-10	Empire of the Petal Throne
11-15	ACKS		11-12	Call of Cthulhu
16	Gamma World		13-14	Dreamlands
17	Metamorphosis Alpha		15	Space 1889
18	(1-3) Runequest (4-6) Warhammer FRP		16	(1-3) Barsoom (4-6) Dark Eye
19-20	Other		17	Traveller
			18	Carcosa
			19-20	Infernum

    This time I'm going to use the technique from ACKS by dividing the encounters up types, in this case movement type.  Swimmers, any creature that can survive an indefinite period in the water; Flyers and Vessels, any object propelled across or under the surface of the water - these range from orcs using a log to swim across the river, to sailing ships to the Apparatus of Kwalish.  Swimmers will be 40% of the encounters, Flyers 10% and Vessels 50%.  (OK, I started here, but ended up adjusting it to take into account ocean creatures swimming further up the estuary, and the Death Tree, which tosses exploding cherries at you, couldn't resist invoking the Rule of Cool.) Any Swimmer that can dive under the water has the choice to avoid the encounter completely, so while 90% of the creatures in the estuary are Swimmers, many of them won't be seen.

  There are three encounter areas on the Roglaroon.
    Mermist Tail (Hexes 2623, 2624,2625, 2726)  From the City State to bend west of Caravan Crossing, this area dominated by the inflow of the Conqueror River and the waters leaching out of the Mermist Swamp.  The estuary changes width rapidly along this stretch. Saltwater species are rarely encountered in this area.

  Dearthwood Reach (Hexes 2825, 2925, 3024, 3124, 3223, 3323, 3422,3522)  The long straight stretch between the bend and the River Hagrost.  The water becomes increasingly saline towards the Hagrost end.  Saltwater species frequently are encountered in this area.

  Roglaroon Proper, aka The Gullet (Hexes 3522, 3521, 3520, 3519, 3518, 3517, 3516, 3615, 3614, 3613, 3713, 3712)  From the River Hagrost to the Mouth, the estuary is a body of salt water inhabited by ocean creatures.

Aquatic Creatures mentioned in The Guide to the City State of the Invincible Overlord, Booklet I, Installment J and Modron.
   Gods:
     Ran the Cruel, Goddess of Death CSIO p55
     Sea Goddess (Unnamed, so I assume Modron), CSIO p14
     Manannan - CSIO p33
     Modron, Goddess of Rivers, Modron p5

Monsters:
    Maelstrom, Giant Sea Snake works for the City State, Modron p 6
    Giant Octopus, CSIO p33
    Giant Leech, CSIO p28
    Giant Shark, Modron p26

Sophonts
    Kopoacinths, CSIO p 10
    Aquatic Elves, CSIO p17
    Mermen, CSIO p28, p36, p38
    Tritons, CSIO p28
    Sea Hag, CSIO  p38, Modron p5
    Sahuagin,  Modron p26

d20	Mermist Tail	Dearthwood Reach	Roglaroon Proper
01	AD&D Shedu	AD&D Kopoacinth	ACKS Sahuagin
02	AD&D Man, Merchant	ACKS NPC Party	AD&D Whale, Small
03	AD&D Man, Merchant	AD&D Man, Merchant	AD&D Man, Merchant
04	AD&D Man, Merchant	Gamma World Keeshin	AD&D Elasmosaurus
05	AD&D Gar, Giant	Runequest Fishermen	AD&D Man, Merchant
06	AD&D Sea Hag	AD&D Man, Merchant	AD&D Man, Pirate  (Orcs in small craft)
07	Warhammer Amoeba	Roglaroon Proper	AD&D Man, Merchant
08	AD&D Owl, Giant	Roglaroon Proper	Gamma World Menarl
09	AD&D Beaver, Giant	AD&D Man, Pirate	AD&D Triton
10	AD&D Man, Buccaneer	AD&D Otter, Giant	ACKS Pegasus
11	AD&D Nixie	AD&D Man, Warship	Dark Eye Krakenewt
12	AD&D Man, Buccaneer	ACKS Lizard Man	AD&D Octopus, Giant
13	ACKS Thief	ACKS Roc, Small	AD&D Man, Merchant
14	Warhammer Chameoleech*	AD&D Hippopotomus	AD&D Man, Buccaneer
15	Roglaroon Proper	Roglaroon Proper	Call of Cthulhu Shantak
16	AD&D Kaolinth	AD&D Man, Merchant	AD&D Shark, Giant
17	AD&D Man, Warship	Roglaroon Proper	AD&D Crocodile, Giant
18	AD&D Gar, Giant	Call of Cthulhu  Nightguants	AD&D Elf, Aquatic
19	Empire of the Petal Throne  Qu'uni	Metamorphosis Alpha Death Tree	ACKS Sea Dragon ("Maelstrom")
20	Infernum Malcubi	AD&D Merman	AD&D Man, Warship

Flyers are italicized

Roglaroon Proper - re-roll on that column for the encounter

* The Chameoleech is best described as a cross between a Giant Leech and a Mimic.

Mostly this is as I rolled it using the appropriate random tables.  Had to make some adjustments to bring the flyers in at the right numbers on the Mermist Tail, added in the Aquatic Elves, Sahuagin and Sea Hag from the canon, and I took the AD&D entry "Man, Pirate (Tribesmen with small craft)" and decided that that would represent Orcs of the Purple Claw.

CSIO - Troubles with the Roglaroon

Estuary of Roglaroon

     I have issues with the Estuary of Roglaroon, geologic and economic issues with it.

     Geographically, the estuary describes a shallow U, short on the left side where it bends through the edge of Dearthwood  past the Mermist Swamp, longer and more angled on the right from River Hagrost to the mouth.  In between, despite some jagged edges, it's an almost straight line, no bends or meanders for some forty miles at the original 5 mile hex scale, 120 miles at the 15 mile hex scale I have seen some substitute.  And it's incredibly uniform in width during that stretch, maybe 2 miles across at the start and three miles at the very end.
     What could cut such a uniform channel?  Normal stream erosion wanders as the water flows through soil and rock of varying hardness, producing a typically serpentine shape of a river.  The landscape also produces significant changes in the river's width, passing through softer, more erosion prone soils to produce riverine lakes, like Lake Pepin on the Mississippi.   Glaciers cut narrow channels when they encounter hard rock, producing fjords, but the banks of the Roglaroon aren't mountainous.  I'm left with considering a glacial outburst flood as the agent to carve the channel.  If we examine the area bounded by Ered Losthian, the Cloudwall Mountains, Quean's Waste and Ered Cantreff in the south and the Howling Hills and the hills of Dearthwood in the north; we can posit a small glacial lake of over 7,000 square miles in extent, with a moraine dam stretching between the hills just east of Caravan Crossing.  Canonically, Wilderlands is a post-Ice Age world, this gives a plausible reasoning for the strange geography of the estuary.
     Related, how far up stream does the saline effect the water?  I would argue that based on the geography, the waters of the Roglaroon are basically salt water all the way up to the Hagrost, with water of diminishing brackishness from there to the City State.  Fortunately for the City State, it gets it's drinking water from an aqueduct out of Oracle Lake, which Groaning Falls protects from contamination by storm surge.

          Moving to economics, I've been having trouble imagining how ships get 140 miles up the estuary  to the City State, it appears so narrow in scale that I didn't think sailing ships would have room to tack across the channel.  However, digging into the detailed information in the Chivalry and Sorcery supplement Bireme and Galley,* yes, caravels at least are nimble enough to tack back and forth across the current and are approximately the size of the ships shown on the reprints of the City State maps.
     This of course led me to crunch numbers to determine how long it would take a ship to sail from the City State to Modron.   There's a lot of factors involved, more with sailing ships than I had worry about driving steam ships.  Sometimes the current fights you, sometimes it accelerates you and of course a tidal current is never constant.   Sometimes the flood is so strong that you have to anchor for a few hours or be carried backwards.  In general, your velocity made good on course (VMC), and hence distance traveled, is the cosine of the angle between the course you are steering and the tack you are on, times your true speed over ground.   Because I need to take the relationships between tide, wind and heading into account, I can run it in a spreadsheet (or write code functions if I feel like it), but haven't been able to reduce it to a simple formula yet.
  So, assuming the maximum tidal current in the Roglaroon is 5 knots; the ship departs the City State at the last of the flood tide; and the ship's true speed over ground equals nominal speed from Bireme and Galley based on the wind strength.
  The fastest time is a mere 9 hours in a Strong Northwest Wind.
  The slowest time is 60.5 hours in a Light Northeast Breeze.

Breakdown:
   Five knot tidal current.  The Thames in London runs 4-6 knots.
   Departing on the last of the flood tide, just before slack water means the ship will ride it's anchor cable to face downstream, reducing any maneuvering to leave port.
   Nominal speeds based on wind strength from Bireme & Galley for a Caravel
            Light - 3.5 knots
            Strong Wind - 9.5 knots

 Bireme and Galley doesn't list actual wind speeds, so my estimation of their definition are based on the Beaufort scale.

Light Breeze    Beaufort Scale 2-3      4-10 knots
Good Breeze    Beaufort Scale 4         11-16 knots

Good Wind       Beaufort Scale 5         17-21 knots

Strong Wind     Beaufort Scale 6         22-27 knots

Average Voyage - Overall 27.1 hours

Average Voyage - Light Breeze   38.8 hours

Average Voyage - Strong Wind   15.3 hours

Predicted Voyage - Light Breeze at true speed over ground  28.5 hours 

Predicted Voyage - Strong Wind at true speed over ground  10.5 hours

* Looks like the pictrues have disappeared from the Bireme and Galley post

Meet the Party - C&S in CSIO

Werebear illustration from AD&D
Monster Manual. 

     I want to use some of the background posts I've been doing in the Wilderlands to run some soloplay Chivalry & Sorcery.  I've already detailed creating the Power Word Magician Mordacris in C&S.  Here's the rest of the party, I generated them using my C&S 1e character generator spreadsheet - which did identify an issue in how it handles Bastardy.

   

Ysolt

Human Female	Age 20			Level 1		XP	0
5’ 5”	110 Dr

Characteristics

DEX	13	ST	20(2)	CON	19	PA	15	BV	17
IQ	16	WIS	19	CHA	21	ALGN	5

Combat

Body	21	Fatigue	9	Carry Cap	638Dr
Military Ability	10.9	Command Level	6.45	PCF	14.33
	Blows	Damage	%HIt	%Parry
Light Weapons	2	3*WDF	+20%	-20%
Heavy Weapons	1	4*WDF	+18%	-12%
% Shield Parry	+18%	Dodge	-22%

Social

     Daughter of Simon, a Freehold Farmer of Tharbian background owing fealty to Baron Jaquob. She is his fifth child and is considered "Good Child". Her Superhuman strength makes her a formidable fighter.  Her dexterity, appearance, bardic voice, intelligence and wisdom - combined with her impressive strength give her a truly commanding presence. She has black hair, bronzed skin and light blue eyes.
She is a Neutrally aspected Leo and gets no bonus to experience.
Her father is a regular supplier of herbs and other spell components to Mordacris' master Athelbur. She has been hired to escort Mordacris to collect some rare spell components on the far side of Dearthwood.

Social Status

8

Basic Influence Factor

18.5

Languages: Common, Tharbian

Useful Skills:

Local Knowledge - identification of crops and livestock common in the area, as well as their state of health. Similarly, she is aware of types of major predators and they're preferred prey. She knows of the major locations around the City State, but has never visited most of them.

_________________________________________________________________________________

Wulfgar

Human Male	Age 22			Level 1		XP	0
5’ 7”	130 Dr

Characteristics

DEX	7	ST	14	CON	12	PA	11	BV	15
IQ	17	WIS	9	CHA	12	ALGN	20

Combat

Body	10	Fatigue	6	Carry Cap	221Dr
Military Ability	3.9	Command Level	1.95	PCF	5.66
	Blows	Damage	%HIt	%Parry
Light Weapons	-	3*WDF	+5%	-5%
Heavy Weapons	_	3*WDF	-	-
% Shield Parry	-3%	Dodge	-5%

Social

Son of Arthur, a peddlar in the City State, he is the 6th child and is considered a "Credit to the Family".  He is an apprentice Assassin in the Thieves Guild, providing muscle and look out services for their heists. He is a poorly aspected Aquarius and suffers a -5% experience penalty. As an Assassin, he only gets 1/2 experience from using his thief skills, but full experience for fighting. He has black hair, olive skin and green-brown eyes. He is a childhood friend of Mordacris and is looking for some time out of town until the Watch forgets his face.

Social Status

2

Basic Influence Factor

8

  

Languages: Common

Useful Skills:
Thief Skills

Detect Trap +35%

Disarm Trap +5%Pick Pocket +5%

Chance of Discovery 45%/10% (at Night)

Local Knowledge - intimately familiar with the streets of the City State, it's markets, shops and temples. He has a 40% chance of finding his way across roof tops to any destination in the city and a 15% of finding his way through the catacombs underneath. He knows a number thieves in the Guild, but does NOT know of the ancient Master Thieves in the catacombs underneath the guildhouse. He can dispose of stolen goods and any items found on adventure through the Thieves Guild.


________________________________________________________________________________

Adenin

Werebear Male	Age 17			Level 1		XP	0
6’ 0”	180 Dr			MKL 1

Characteristics (Human)

DEX	1	ST	8	CON	10	PA	20	BV	17
IQ	17	WIS	20	CHA	20/22/14	ALGN	7

Combat

Body	11	Fatigue	3	Carry Cap	144Dr
Military Ability	8.8	Command Level	5.4	PCF	6.54
	Blows	Damage	%HIt	%Parry
Light Weapons	-	3*WDF	+8%	-8%
Heavy Weapons	-	3*WDF	+5%	-
% Shield Parry	-	Dodge	-8%

Social

     Baron Jaquob was an avid hunter in his youth. Once while out hunting, he spent an afternoon of dalliance with the daughter of a woodcutter. The following summer, he slew a sow bear and captured it's single cub. Upon death, the bear transformed back into the form of the woodcutter's daughter.  When the moon waned, the cub changed into a beautiful baby boy.
     The baron has never acknowledged the boy's paternity, but has raised him as his ward in his household. Adenin's clumsiness and middling strength, combined with his stellar intellect has caused his protector to obtain a place for him as a Priest of Uller. He has been sent to the Temple of Odin in the City State for training. His god-like appearance gives him a 22 Charisma with members of the opposite sex, 20 Charisma with members of the same sex and a 14 Charisma when dealing with members of either sex who are Ugly. (Those unfortunates with a Personal Appearance of less than 8.) He is a poorly aspected Aquarius. He receives +5% experience as a Cleric, -5% experience as a Fighter and a Magic User.

     As part of his training, he has been assigned to spend three weeks in an isolated cell in the eastern fringes of the Dearthwood. He will travel there with Ysolt and Mordacris. He has auburn hair, gray eyes and his complexion is a light yellow (Hex #FAFAD2 if you're having trouble visualizing it.)

Note: The spreadsheet didn't show him as an unacknowledged Bastard, and didn't calculate the Social Status correctly.  But the error gave me the idea of having him be the ward of his biological father.  And the rest is backstory.

Social Status

10

Basic Influence Factor

21

  Magick

Clerical Miracles Number/day: 1

Miracles Known

Cure Minor Wounds (6-10 point restored to Body and Fatigue. May be received 1/day by any character)
Note: He had a 60% chance of also having Purify Food & Water, but rolled too high. He can try again when he achieves 2nd level.

Miraculous Magical Effects

     All Clerics have a 60% - 5% per Alignment over 1 to be able to cast spells as a Primitive Talent Magic User, again he rolled too high. Clerics with Intelligence, Wisdom and Charisma of 15+, have the same percentage chance cast spells as a Shaman. He rolled a 28, the three weeks in the cell is when he will encounter his Totem Spirit (predetermined to be a bear). Until then he cannot cast spells.

Concentration Level

3.9

Personal Magick Factor

9.8

Magick Level

1

Spell Level

1

Direct Injurious Effect	Volume Multiplier (Missiles)	Volume Multiplier (Material)	Time Factor	Spell Range
2*EDF	2*	5*	4*MKL	10’ *MKL

Languages: Common, Tharbian

Spells:

Useful Skills:
Multiclass - Raised as a noble, he can fight as a Knight. He can earn experience as a fighter, cleric or magic user. Any experience will be based on the most favorable way of calculating it.

Local Knowledge - identification of crops and livestock common in the area. Similarly, he is aware of types of hunting prey and they're preferred habitats. He knows of the major locations around the City State, but has never visited most of them. He knows of major nobility in the City State and can correctly identify their heraldry and major manor locations 90% of the time. He is not personally acquainted with them.


Characteristics (Bear)
Body 70 Weight 900Dr Move 60/180

Armor 5 Hit +35% Dodge -25%

4*WDF MLC 6 Claws - 3 attacks, 14 points of damage each. H Weapon Bash
2*WDF MLB 3 Fangs - 2 attacks, 8 points damage each. L* Weapon Bash.
Hug - costs one blow. 3-18 damage (AC less than 6) 2-12 damage= AC greater than 5). May toss for 5-30 feet with a 25% chance of the victim being Stunned.