Probability

[Ravendas]

One thing I love about games is figuring out the chances of things happening. So I made a thread on a different forum to post about probability things I have done, but it has fallen into disuse, and is pretty cluttered, so I thought I'd repost a bit of it over here, to perhaps start a bit of discussion.

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Advance Rolls
The 2d6 chart on page 87 of the ORB has lots of things going on, but which things are common, which are uncommon? Well, percentage wise, there are only 4 brackets. Here they are:
27.7% --> New skill, based on gang type.
13.8% --> WS, BS
8.3% --> I, LD
5.5% --> S, T, W, A, or Any new skill.

So every time a character levels up, he has about a 1/3 chance of getting a skill. The hard ones to get are some of the best, like Toughness and Wounds, which makes sense.

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Red Dot Laser Sight
This sight gives the user a +1 chance to hit, but gives the person you hit a 1/6 chance to dodge it. Originally my old group thought "That's dumb, gives you +1/6 chance to hit, and then they ignore it 1/6 of the time, it cancels itself out!"
Well, not really. Not all 1/6 are equal.

This little chart has 4 columns.
Roll Req - Number someone would need to hit their target, without the laser sight.
Base% -  Percent chance to hit, without the sight.
Dot% -  Chance of hitting with the sight (taking the dodge roll into account). This is ((Base% + 16.7%) X 83.3%)
%Improvement - How much better the dot% is than the base%. This is found by (Dot% / Base%).

Code: [Select]

Roll Req    Base%    Dot%    %Improvement
9+          2.8%     4.6%    64.3%
8+          5.6%     6.9%    23.2%
7+          8.3%     13.9%   67.5%
6+          16.7%    27.7%   65.8%
5+          33.3%    41.7%   25.2%
4+          50%      55.6%   11.1%
3+          66.7%    69.4%   4%
Rolls of 2 are actually hurt by the dot, since there is no improved chance to hit, it would just give them a dodge save, so you would turn it off in that case.

So what this means is that, the worse your chance to hit, the more it helps you hit. So despite what your first impulse is, don't give the dot to a sniper type, give it to that juve or ganger that can't seem to hit the broad side of a barn. They will get much more use out of it.

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Archeotech Territory, and the chance of abusing and losing it

So, if you have archeotech, you can get 2d6x10 from it with no worries, or bump it up to 3,4,5 or 6 dice. Thing is, roll a double, and you lose the archeotech. So what is the chance of messing it up for each one?

Roll 3 dice -> 1-(1 x 5/6 x 4/6) =1-(20/36) -->  44.4% Chance of losing.
Roll 4 dice -> 1-(1 x 5/6 x 4/6 x 3/6) = 1-(60/216) --> 72.2% Chance of losing.
Roll 5 dice -> (math omitted, gets long) --> 90.8% Chance of losing.
Roll 6 dice -> (this is equivalent to rolling a Yahtzee, no rerolls) --> 98.46% Chance of losing.

For some explanation of how the math works, I'll explain the 3 dice roll, since it just expands out depending on the number of dice.

You just figure out the chance of it not happening, aka you roll 3 separate numbers, and then subtract the one. So, the first dice you roll has a 100% chance of being a unique number, since it is the first dice. Then, the 2nd dice has a 5/6 chance of being a unique number, any number besides what the first dice is. The 3rd dice has a 4/6 chance of being a unique number, any number besides what the first or second dice were. Therefore, you have a 1 x 5/6 x 4/6 chance of passing the test, or a 1 - (1 x 5/6 x 4/6) chance of failing it.

So yeah, even rolling 3 dice on it is asking for a good chance of failing. 5 or more and passing becomes a miracle. The fixer trait lets you reroll, making it so you have to fail the test twice. This is just (Chance of losing)^2.

Fixer results would be:
3 dice --> 19.7% Chance of losing
4 dice --> 52.1% Chance of losing
5 dice --> 82.4% Chance of losing
6 dice --> 96.9% Chance of losing

So fixers make 3 or 4 dice more manageable, but anything above is definitely reckless.

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Mung Vase

These things are just a gamble, but how good of a gamble? What's your chance of profiting from this? Well, lets look at the numbers.

The cost: 10 X 1d6. 10 creds at the cheapest, 60 at the most. Average cost of 35 creds

For the sales values, you roll a d6. Some are terrible, some are actually pretty good.
1: Fake, worthless, 0 creds.
2: "Fake", but worth 1d6 creds. Low - 1, High - 6, Avg - 3.5
3: 30+4d6 creds. Low - 34, High - 54, Avg - 44
4: 30+6d6 creds. Low - 36, High - 66, Avg - 51
5: 5x2d6 creds. Low - 10, High - 60, Avg - 35 (this is worse than the 3rd or 4th outcomes!)
6: 10x2d6 creds. Low - 20, High - 120, Avg - 70

So, if the average price of the vase is 35, then on a 3,4 and 6 you will on average make some money, on a 5 you will break even, and a 1 or 2 you will lose badly.

The overall average sale price is: (sum of averages / total outcomes) = (0+3.5+44+51+35+70) / 6 = ~33.91 creds.

On average, if you roll up a mung vase, you will probably lose money. But if that vase is only 10 or 20 credits, I would definitely advise snatching it up. If your vase is offered at 50 or 60, definitely pass. If you brought along a ganger to help you search for items... well... lets just hope you roll low for the buying price. If you are insanely lucky, you could make 110 credits out of this deal. On the opposite end, you could just throw 60 credits into the nearest chempit and come out equal.

[Ravendas]

Average income of a standard gang owning 5 territories [/u]

Basically: Average income from every territory X the chance of having every territory, X 5.
(These are mostly for me, it's average outcome X chance of having the territory)
Chempit - 0.1944
Old Ruins - 1.3888
Slag - 2.0833
Mineral Outcrop - 0.9722
Settlement - 4.1666
Mine Workings - 0.9722
Tunnels - 0.5555
Vents - 0.5555
Holesteads - 1.944
Water Still - 1.944
Drinking Hole - 1.944
Guilder Contact - 1.944
Friendly Doc - 0.9722
Workshop - 0.9722
Gambling Den - 46.6666 / 36 = 1.2962
Spore Cave - 1.944
Archeotech - 1.944
Green Hivers (treated as best, 2d6x10) - 1.944

Total: 22.5983 Creds
That is the average income from all of the various types in the game, weighted by the chance they are rolled up.

22.5983 X 5 territories = ~113 income.

Go across the chart will give you this: (code for formatting)

Code: [Select]

INCOME 1-3 4-6 7-9 10-12 13-15 16-18 19-21
80-119  50 45  40   30    20     5     0

-------------------------------------------
Scare Gas Grenade Effectiveness!

(Another member of the forum, Pyright, helped fix a little mistake in this one, thanks again!)

Scare gas grenades require a model to roll equal to or higher than their toughness and a leadership test to break and run away. I will do this in 2 charts, one for toughness 3, one for toughness 4, since those are the two main toughness numbers used. If someone has a chest wound, well, whatever.

Chance to fail both tests and run away:
Toughness 3:
LD6 - 38.89%
LD7 - 27.78%
LD8 - 18.52%
LD9 - 11.11%

Toughness 4:
LD6 - 29.16%
LD7 - 20.83%
LD8 - 13.88%
LD9 - 8.33%

But the thing is, gas clouds typically hang around for awhile. So most people hit by a gas grenade will actually get hit twice by it, once on the turn thrown, once on their own turn. Only on a 1 does the gas go away completely.

Chance of Succumbing to Scare Gas, when hit twice by it, which will most likely happen (the hitting part, not the succumbing part).
The chance of passing both tests would be (1-failchance)^2. The chance of failing at least one would be 1-(1-failchance)^2.
With the long title out of the way, and the math explained above, here we go.

Chance to fail the toughness and leadership tests at least once
Toughness 3:
LD6 - 62.66%
LD7 - 47.84%
LD8 - 33.61%
LD9 - 20.99%

Toughness 4:
LD6 - 49.82%
LD7 - 37.33%
LD8 - 25.84%
LD9 - 15.97%

This makes them seem a bit more credible. Hit a group of juves, and they will be losing 2/3 of their pack. Even base gangers are almost 50/50 on running away.

And just another thing that came out of this:

Chance to Pass or Fail a Leadership Test
Pass:
LD6 - 41.66%
LD7 - 58.33%
LD8 - 72.22%
LD9 - 83.33%
(LD9 is the same chance as rolling a 2+ on a single die)

Fail: (just 1-pass)
LD6 - 58.33%
LD7 - 41.66%
LD8 - 27.78%
LD9 - 16.67%

For those gangs with a gang leader that has both 9 leadership and Iron Will, you only have a 1/36 chance of involuntarily bottling, or a 2.8% chance.

[Ravendas]

Okay, that is all the old stuff I posted before, collected together into a few posts in a row. Anyone have any other problems they would like worked out? Or any questions/comments/etc about the above?

I did the math for the Muscle skill Headbutt, and it turns out it's basically worse using it than not using it in 99% of situations. I have the math written down on a piece of paper somewhere, I just gotta find it. I'll post it here later.

For me at least, doing a lot of this and figuring things out changed the way I thought about a lot of things. I was originally annoyed when a juve got +ld, but now that I see that the jump from 6 to 7 leadership is so huge, I'm always hoping for it. The red dot laser sight math made me actually want to buy it as well, since when I first started playing, and hardly knew anything about probability, figured it was worthless. And the archeotech math made me too scared to ever roll more than 2 dice, since it's such bad odds to lose such an awesome territory.

[ANSWER_MOD_DABANK]

Great Info. Keep it up.

[Ravendas]

Headbutt Comparison

Okay, I always had a feeling the basic rules for headbutt were terrible, and now I have math to back it up.

The rules for headbutt are you can trade in all your hits you made on an enemy for 1 big hit, at +1 strength per additional hit. So a normal strength 3 model would hit at strength 4 if he made two hits, strength 5 with 3 hits, etc.

I made 4 charts, each one representing a different strength/toughness comparison. The first one is Toughness-1, meaning the attacker has 1 more strength than his opponent. 4S/3T, or 5S/4T, etc.

The rows are the number of hits the attacker scored, the columns are the chance to get X wounds based on the number of hits. The column 'Chance +1 Wounds' is just the sum of all the previous columns. That is the important number, the chance to score at least 1 wound on the enemy. Red boxes mean it's impossible to get that result. Finally the last column is just a reminder, so it's easier to see what is doing what percentage. The green name is the one that is better in the given situation.

I didn't do a Toughness -2 chart, because in every case it is better to just use your normal hits, and not headbutt. Headbutt gives no bonus there.

Now, without further ado, Toughness -1 (3S/2T, 4S/3T, etc):


Taking your hits normally is better than headbutt in every situation.

Equal Toughness (3S/3T, 4S/4T, etc):


Once again, headbutt is worse every time

Toughness +1 (3S/4T, 4S/5T, etc):


Headbutt is better in two situations, and just barely. If you are fighting someone and get 4 or 5 hits on them, and they have 1 more toughness, then headbutt them. Unless you want the chance of doing more than 1 wound that is...

Toughness +2 (3S/5T, 4S/6T, etc):


Finally, headbutt is useful. Headbutt beats the normal hits every single time here. However, how often does a fighter try to hit something with 2 more toughness than his strength? Even then, headbutt has no chance of dealing more than 1 wound, so if your enemy has multiple wounds, headbutting might not finish the job.

If anyone has any questions about this math, let me know. If you redo my math and are off by a few tenths of a percent here or there, it's because I was using my cellphone calculator for most of this, and it loved to say 'Syntax Error' with the longer calculations.

[Madness]

I redid part of the maths, check it on google docs, skip to the third sheet (bottom, click on headbutt stats)

The math is solid, we got the same results, BUT we completely ignored the armor issue (higher str also grants a higher armor negative modifier) a quick calculation 2 hits str 3 versus armor 4+ without headbutt:
4+ to wound (50%) * 3- to fail the save (50%) = 25% to wound with the first hit
(chances that I need the second hit to wound once (75%) * 25%) + chances to wound with the first hit (25%) = 43.75%

With headbutt:
3+ to wound (66.67%) * 4- to fail the save (66.67%)  = 44.44%

The higher the armor value gets, the lower the strength of the original attack is, the worse the original armor mod was the better headbutt gets at scoring that first hit.

But it's really underwhelming. Maybe add the option to use every traded attack to cause one more wound or add +1 to Str is more interesting.

[Ravendas]

Yeah, you have a good point with the armor. The thing is, armor is extremely rare in Necromunda, so it didn't even come to me to account for it.

SkillsV2 changed it so that it has the same effect as normal, plus if the headbuttee survives, his weaponskill is halved the next round of combat, due to being stunned. This actually makes it very useful, as you will have a much better chance of winning and finishing the person off. Luckily I just restarted my little campaign, and we are switching over to the 'official' test SkillsV2.

Edit: \/\/ http://www.sg.tacticalwargames.net/forum/index.php?topic=268.0 \/\/

[Madness]

Um, where are the skills v2?

[Ravendas]

Just remembered I posted this somewhere in a Warseer thread. Someone was arguing that Stubguns are useful in the hands of a juve, I did this to compare Autopistols and Stubguns.

Stubguns with Dumdums vs Autopistols, the legendary battle.
Unless noted otherwise, these are against 3 toughness targets. For brevity, AP is autopistol, SG is stubgun.
Stats:
AP, +2 short/- Long, str3
SG, - Short/-1 Long, str4, Explodes if an ammo roll is failed. If it explodes, it is a str3 hit on the juve, wounding him on a 4+.

Hit % is the chance to hit the target in the given situation, Hit+Wound is the chance you will both hit and wound the target. Both numbers are important, as the first one is the chance to pin the target, and the 2nd one is the chance to actually do some wound causing damage.

2BS, LONG RANGE

No Cover AP - Hit 33.3%, Hit+Wound 16.7%
No Cover SG - Hit 16.7%, Hit+Wound 11.1%, Chance to Explode on a Hit 50%

Light Cover AP - Hit 16.7%, Hit+Wound 8.3%
Light Cover SG - Hit 8.3%, Hit+Wound 5.5%, Chance to Explode on a Hit 50%

Hard Cover AP - Hit 8.3%, Hit+Wound 4.2%
Hard Cover SG - Hit 5.6%, Hit+Wound 3.7%, Chance to Explode on a Hit 50%

2BS, SHORT RANGE

No Cover AP - Hit 66.6%, Hit+Wound 33.3%
No Cover SG - Hit 33.3%, Hit+Wound 22.2%, Chance to Explode on a Hit 25%

Light Cover AP - Hit 50.0%, Hit+Wound 25%
Light Cover SG - Hit 16.7%, Hit+Wound 11.1%, Chance to Explode on a Hit 50%

Hard Cover AP - Hit 33.3%, Hit+Wound 16.7%
Hard Cover SG - Hit 8.3%, Hit+Wound 5.5%, Chance to Explode on a Hit 50%

In every situation, the Autopistol has both a higher chance to hit, and a higher chance to wound, despite having 1 less strength than the Stubgun. Also, using the autopistol is much safer, due to it exploding 8.33% of the time you are required to take an ammo check, and only being a strength 2 explosion.

And yes, I didn't include the stubguns -1 armor, because armor is so rarely used in Necromunda, and I didn't feel like complicating the numbers further for something that so rarely comes into play.

[Madness]

We need some mathematical formula to be quickly copypasted for statistics. Any math major around?

[Ant]

I'm a bit of a numpty concerning maths so you won't see my face in this topic very often (btw, ooh colourful graphs!), but I just thought I'd pop in and say what a great resource this topic is. Great job Rav. 

[Ravendas]

I'm a bit of a numpty concerning maths so you won't see my face in this topic very often (btw, ooh colourful graphs!), but I just thought I'd pop in and say what a great resource this topic is. Great job Rav. 

Thanks! You should see my rulebook, full of little notes like this all over. Despite all my math stuff, I actually don't min-max my gangs when I play. I just like to know the chances of things happening, so if something crazy happens, I know just how crazy it is.

And Madness, I sadly only have a math minor so I guess I don't qualify for whatever it is that you need. Care to elucidate?

[Madness]

It's pretty simple actually, we need some formulas who still retain variables in variable form to be used in complex calculations. For instance, take one of the easiest things to calculate, the chance to hit with a weapon.

You have a variable called Adjusted BS with an aBS of 3 you have 3/6 chances to hit, or 50% in percent form. The formula is pretty simple, you have 7-3 (as per book) which is the least amount you need to obtain with the die. aBS 3->4+ aBS 5->2+. Then we have to calculate the chances of getting a 4+, which are 1-((4-1)/6) (the inverse of the chances of getting 3-).

The resulting formula is 1-(((7-aBS)-1)/6) which is quickly simplified in aBS/6. BUT this is only valid for aBS>0 and aBS<6, because for aBS 0 it's 1/6 * 3/6, aBS -1 1/6 * 2/6, aBS -2 1/6 * 1/6. But this is hardly a formulaic version.

The benefit of having it all with parameters is that we could calculate very complex stuff without breaking it down in multiple cases, it would be nice to know exactly what strength bonus is necessary to offset the parry in a fight with 1 attack and equivalent WS, but it would be also nice to be able to alter one parameter and see the equation being balanced once again.

So, summon your nerd friends and let's get dirty.

[Ravendas]

Forgot about this one from awhile back. This is showing the differences in chance to hit, between a typical weapon and a flail in hand to hand.

Using these modifiers in my script, this is what I came up with.
A: Every roll of 1 is a -1 modifier to your highest roll (critical fumble technically adds +1 to your enemy's, but this is the same thing).

B: Every roll of 6 beyond the first is a +1 modifier to your highest roll.

Here is what I got for attack dice, from 1A to 4A.
There are 3 columns. The first lists the outcome, from 0 or less, to 7 or greater. I could find out individual numbers, but figure that condensing the overkills and crazy fumbles would make the numbers easier to read.

FREQ is frequency, how many times that outcome comes up when rolling that many dice.
PERC is the percentage chance you will get that outcome, based on FREQ/Total. This is the important number
At the end of each listing is "Average throw". This is also important, as this is what you will get in an average fight using this many dice.

Code: [Select]

One Dice Rolled:
FREQ                 PERC
0- = 1                16.66666%
1  = 0                0%
2  = 1                16.66666%
3  = 1                16.66666%
4  = 1                16.66666%
5  = 1                16.66666%
6  = 1                16.66666%
7+ = 0                0%
Total Throws: 6
Average throw: 3.33

-----------------------------------------

Two Dice Rolled:
FREQ           PERC
0- = 1          2.78%
1 = 2          5.56%
2 = 3          8.33%
3 = 5          13.89%
4 = 7          19.44%
5 = 9          25%
6 = 8          22.22%
7+ = 1          2.78%
Total throws: 36
Average throw: 4.17

-----------------------------------------

Three Dice Rolled:
FREQ           PERC
0- = 4         1.85%
1 = 6          2.78%
2 = 13          6.02%
3 = 25          11.57%
4 = 43          19.91%
5 = 61          28.24%
6 = 51          23.61%
7+ = 13         6.02%
Total throws: 216
Average throw: 4.54

-----------------------------------------

Four Dice Rolled:
FREQ           PERC
0- = 15         1.16%
1 = 26          2.01%
2 = 63          4.86%
3 = 137          10.57%
4 = 261          20.14%
5 = 373          28.78%
6 = 304          23.46%
7+ = 117         9.03%
Total throws: 1296
Average throw: 4.73

This has an obvious upwards trend, where more dice is better. The chance to roll a 7+ increases steadily, while the 0- drops off quickly. But what about when you use a flail? Flails double the fumble penalty. Will it still have the same trend we see here? Lets see:

Outcomes when using a chain or flail

Code: [Select]

One Attack Dice: (Using a chain/flail)
    FREQ        PERC
0- = 1          16.6%
1 = 0           0%
2 = 1           16.6%
3 = 1           16.6%
4 = 1           16.6%
5 = 1           16.6%
6 = 1           16.6%
Total throws: 6
Average throw: 3.17

-----------------------------------------

Two Dice Rolled: (using a chain/flail)
FREQ           PERC
0- = 3          8.33%
1 = 2          5.56%
2 = 3          8.33%
3 = 5          13.89%
4 = 7          19.44%
5 = 7          19.44%
6 = 8          22.22%
7+ = 1          2.78%
Total throws: 36
Average throw: 3.83

-----------------------------------------

Three Dice Rolled: (using a chain/flail)
FREQ           PERC
0- = 13         6.02%
1 = 12          5.56%
2 = 19          8.80%
3 = 28          12.96%
4 = 43          19.91%
5 = 40          18.53%
6 = 48          22.22%
7+ = 13         6.02%
Total throws: 216
Average throw: 4.04

-----------------------------------------

Four Dice Rolled: (using a chain/flail)
FREQ           PERC
0- = 79         6.10%
1 = 70          5.40%
2 = 125         9.65%
3 = 169         13.04%
4 = 257         19.83%
5 = 223         17.21%
6 = 260         20.06%
7+ = 113        8.71%
Total throws: 1296
Average throw: 4.06

The chance to hit with a flail is obviously lower than the standard weapon, plus it doesn't have the same trend. Whereas a typical weapon has an upward trend, plateauing at a combined score of 5, flails actually have a bell curve. After 4 dice, the chance to hit with a flail starts to drop off, where the extra dice just give you a greater chance to fumble.

This is concisely shown below. This section was written and done by Pyright from the eastern fringe forums.
Expected Values for Normal Weapons

Code: [Select]

Dice Rolled:  1 | Expected Value: 3.33
Dice Rolled:  2 | Expected Value: 4.17
Dice Rolled:  3 | Expected Value: 4.54
Dice Rolled:  4 | Expected Value: 4.73
Dice Rolled:  5 | Expected Value: 4.83
Dice Rolled:  6 | Expected Value: 4.90
Dice Rolled:  7 | Expected Value: 4.93
Dice Rolled:  8 | Expected Value: 4.96
Dice Rolled:  9 | Expected Value: 4.97
Dice Rolled: 10 | Expected Value: 4.98
Dice Rolled: 11 | Expected Value: 4.99
Dice Rolled: 12 | Expected Value: 4.99
Dice Rolled: 13 | Expected Value: 5.00
Dice Rolled: 14 | Expected Value: 5.00
Dice Rolled: 15 | Expected Value: 5.00
Dice Rolled: 16 | Expected Value: 5.00
Dice Rolled: 17 | Expected Value: 5.00
Dice Rolled: 18 | Expected Value: 5.00
Dice Rolled: 19 | Expected Value: 5.00
Dice Rolled: 20 | Expected Value: 5.00

Expected Value for Chain/Flail

Code: [Select]

Dice Rolled:  1 | Expected Value: 3.17
Dice Rolled:  2 | Expected Value: 3.83
Dice Rolled:  3 | Expected Value: 4.04
Dice Rolled:  4 | Expected Value: 4.06
Dice Rolled:  5 | Expected Value: 4.00
Dice Rolled:  6 | Expected Value: 3.90
Dice Rolled:  7 | Expected Value: 3.77
Dice Rolled:  8 | Expected Value: 3.62
Dice Rolled:  9 | Expected Value: 3.47
Dice Rolled: 10 | Expected Value: 3.32
Dice Rolled: 11 | Expected Value: 3.15
Dice Rolled: 12 | Expected Value: 2.99
Dice Rolled: 13 | Expected Value: 2.83
Dice Rolled: 14 | Expected Value: 2.66
Dice Rolled: 15 | Expected Value: 2.50
Dice Rolled: 16 | Expected Value: 2.33
Dice Rolled: 17 | Expected Value: 2.16
Dice Rolled: 18 | Expected Value: 2.00
Dice Rolled: 19 | Expected Value: 1.83
Dice Rolled: 20 | Expected Value: 1.67

So, to sum up all of this, more attacks is always good for a standard weapon. Eventually it doesn't help very much, and just a straight weapon skill increase is better, but extra attacks always help standard weapons. However for flails, their usefulness caps at 4 dice. Anything more and you really start hamstringing yourself. This is at odds with the typical Warhammer fluff, of flail models having frenzy, increasing their attack dice, which can really hurt when using a chain or flail.

Edit: Fixed a c/p error, chopped of long trailing digits in one part.

[Caelwyn]

Its good stuff but for the 1 dice flail results you have 7 results all listing 16.6% chance. Shouldn't the result of +1 to combat score still be a 0 result.

Its maintained something i've always felt, that the flail definitely isn't a very effective hand to hand weapon. It's interesting to see that after 3 attacks the benefits become smaller and smaller (if i've got it right) for a hand to hand focussed fighter.