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mortiferum
26-04-2007, 12:09
Hi

Trying to work out some complicated stats (atleast for me!)

Which unit is better at glancing or penetrating armour values 12 and 13 at a range of 12”?

EDIT: the "sucess critieria" is simply to obtain a roll on the glance OR penetrating damage tables.
(I'm NOT interested in whether the result is vehicle destroyed, immobilised, etc...)

5 Fire Dragons

OR

3 Wraithguard

Thanks

Morty

Carlos
26-04-2007, 12:54
Impossible to predict, given Dice dont follow the laws of probability, they follow the laws of physics and thermodynamics.

That said your 5 Fire Dragons have MORE SHOTS and thus increase their odds.

Enderel
26-04-2007, 13:03
As Carlos said it's difficult to predict.

But as a rough guide you could look at it this way.

Wraithguard 3 (105pts) hits on 3's. Three shots so on average 2 hits. Of those you have a 66% chance of inflicting a glance or pen shot at anything upto 12"

Fire Dragons 6 (96pts) hits on 3's. So on average 4 hits. At 12" against AV12 50/50 chance of glance or better. At 6" 66% chance of glance or better.

At 12" against AV13 33% chance of glance or better. At 6" 50% (very roughly) chance of glance or better.

Fire dragons for me overall!

Really does depend on luck..

BloodiedSword
26-04-2007, 13:33
Let's just look at it from a "chance to destroy directly" point of view (so the result you want is any Vehicle Destroyed result. Ignore skimmers for now).

The expected number of Vehicle Destroyed results with Wraithguard is

From glancing hits:

3 shots * (2/3 to hit) * (1/3 to glance) * (1/6 vehicle destroyed) = 1/9

From penetrating hits:

3 shots * (2/3 to hit) * (1/3 to penetrate) * (1/2 vehicle destroyed) = 1/3

So in total you expect to get 4/9 of a Vehicle Destroyed result from one volley.

Fire Dragons: (You specified 5 so I'll go with that first)

From glancing hits - impossible as Fusion guns are AP1.

From penetrating hits -

5 shots * (2/3 to hit) * (1/2 to penetrate) * (1/2 vehicle destroyed) = 5/6

So the Fire Dragons expect to cause 5/6 of a Vehicle Destroyed result per volley which is significantly better.

Galley
26-04-2007, 13:35
Pardon me if I sound ignorant here...

But if the Wraithguards have a 66% chance regardless of armor or distance, then I'd venture to say that they're the better choice.

mortiferum
26-04-2007, 13:50
Impossible to predict.

Is this correct? Perhaps I didn't word my thread correctly but surely there must be a statistical (or should I be using the word probability?) calculation that would hightlight the percentages of each unit?

Here's what I have for Wraithguard (though I think my calculations might be flawed?)

http://i34.photobucket.com/albums/d112/r555ung/Internet%20Play/Presentation1.jpg

I haven't yet calculated for Fire Dragons since I'm not totally convincing of my calculations?

The probabilities for the 2nd and 3rd shots being a success takes into account the preceding shots having failed, so for the second shot I've multipled 0.56*0.67*0.67.

Morty

BloodiedSword
26-04-2007, 14:05
You need to quantify what you require more clearly.

I don't think it's reasonable to consider just "causing a penetrating or glancing hit" as a "success", mainly because a Penetrating hit is much more devastating than a glancing hit.

See my post earlier if you want some actual number crunching instead of people going "oh, you can't predict it" or "my gut feeling is this".

EDIT: Also, before you embark on any kind of statistical analysis, you must have a very clear understanding of the difference between two ideas:

1. The probability of an event happening (e.g., the probability of causing at least one glancing or penetrating hit)

and

2. The expected value (or average, in other words) of a random variable (e.g., the average number of glancing or penetrating hits you expect to cause).

mortiferum
26-04-2007, 14:25
You need to quantify what you require more clearly.

I don't think it's reasonable to consider just "causing a penetrating or glancing hit" as a "success", mainly because a Penetrating hit is much more devastating than a glancing hit.

See my post earlier if you want some actual number crunching instead of people going "oh, you can't predict it" or "my gut feeling is this".

EDIT: Also, before you embark on any kind of statistical analysis, you must have a very clear understanding of the difference between two ideas:

1. The probability of an event happening (e.g., the probability of causing at least one glancing or penetrating hit)

and

2. The expected value (or average, in other words) of a random variable (e.g., the average number of glancing or penetrating hits you expect to cause).

Firstly - many thanks for taking the time to read and reply to my post and thank you for the statistical analaysis definitions - I got an E grade for A level Maths (and that was by avoiding all the stats questions) so as you can tell - its not one of my strong points :D

You say I haven't quantified my question well enough? - the "sucess critieria" is simply to obtain a roll on the glance OR penetrating damage tables.

Armour Penetration - For a wraithcannon its a roll of 3+ regardless of AV. For a Fire Dragon its either a 4+ vs AV12 or a 5+ vs AV13 - my difficulties is factoring in no. of shots from the unit since I know / understand you don't just triple the probablity of 1 shot (in the case of the wraithguard unit)

Yes, I understand the differences of effects on the two tables (glance and penetrate) but to include these would become even more complex - since I would ultimately want to factor in the rules for skimmers (I noticed you even avoided this in your example) and wargear items like Eldar - Holofields, Tau - Decoy Launchers etc. As you can see it quickly becomes more complex and difficult to understand (for a non statitician, like myself) therefore, to simplify the calculations I determined that the best 'succcess criteria' is as I stated above.

BloodiedSword
26-04-2007, 14:41
That makes it much simpler then.

The probability of obtaining at least 1 success with 3 Wraithguard is

1 - (chance of them all failing)

The chance of a single Wraithguard causing a success is 2/3 * 2/3 = 4/9 and so the chance of it failing is 5/9. Thus the chance of causing at least one success is

1 - (5/9 * 5/9 * 5/9) = 1 - 125/729 = 604/729

That's over 75% (someone with a calculator can work it out), which is not bad.

The expected number of successes from 3 Wraithguard is just

3 * (chance of individual success) = 3 * 4/9 = 12/9 = 4/3

Meanwhile for the Fire Dragons, chance of individual success is 2/3 * 1/2 = 1/3 (against AV 12). Chance of at least one success is

1 - (chance they all fail) = 1 - (2/3)^5 = 1 - 32/243 = 211/243

Pretty good odds eh. Against AV 13 this falls to

1 - (7/9)^5 = 1 - (urgh, no calculator but somewhere in the region of 16000/60000) which is roughly 4/15. Someone with a calculator work this one out..

The expected number of successes from the Fire Dragons is 5 * 1/3 = 5/3 against AV 12 and 5 * 2/9 = 10/9 against AV13.

EDIT: Summary to make this readable:

CHANCE OF AT LEAST 1 GLANCING/PENETRATING HIT

3 Wraithcannons against any AV: 83%
5 Fusion Guns against AV 12: 87%
5 Fusion Guns against AV 13: 72%

EXPECTED NUMBER OF GLANCING/PENETRATING HITS

3 Wraithcannons against any AV: 1.33
5 Fusion Guns against AV 12: 1.67
5 Fusion Guns against AV 13: 1.11

Bear in mind that half of the Wraithcannon successes are Glancing hits whilst all of the Fusion Gun successes are Penetrating.

Quin 242
26-04-2007, 14:49
Sword, can you toss in an analysis of FD against AV14
with the added option of the Monolith rules?

And for that matter.. with Eldar Wave serpent tech for firing both against?

mortiferum
26-04-2007, 15:02
EDIT: Summary to make this readable:

CHANCE OF AT LEAST 1 GLANCING/PENETRATING HIT

3 Wraithcannons against any AV: 83%
5 Fusion Guns against AV 12: 87%
5 Fusion Guns against AV 13: 72%

EXPECTED NUMBER OF GLANCING/PENETRATING HITS

3 Wraithcannons against any AV: 1.33
5 Fusion Guns against AV 12: 1.67
5 Fusion Guns against AV 13: 1.11

Hey thanks very much!

Its exactly the same probabilities I generated using my template above - though yours is simplier to calculate 1 - (chance of all failing)

for the person asking for stats on AV14
CHANCE OF AT LEAST 1 GLANCING/PENETRATING HIT
5 Fusion Guns against AV 14: 45%

BloodiedSword
26-04-2007, 15:02
Sword, can you toss in an analysis of FD against AV14
with the added option of the Monolith rules?

And for that matter.. with Eldar Wave serpent tech for firing both against?

All the analysis so far has been from 12" away (i.e., not even using the Melta rules).

For AV14, using 1D6 pen:

Chance of at least one penetrating hit is 45% with 5 Fusion guns.
Expected number of Penetrating hits is 0.56

For AV14, using 2D6 pen:

Chance of at least one penetrating hit is 96% (!!!) with 5 Fusion guns
Expected number of Penetrating hits is 65/27 which is roughly 2.5

As you can imagine, for lower AVs if the Fusion guns get into range they get even more efficient.

I can't remember what the current Wave Serpent rules are.. is it re-rolling successful penetration rolls?

Quin 242
26-04-2007, 15:07
Any weapon over 12 goes down to 12?
and no extra dice for meltas I belive... Sheesh I might have to look that up...

And for the monolith, no extra melta dice.

mortiferum
26-04-2007, 15:08
I can't remember what the current Wave Serpent rules are.. is it re-rolling successful penetration rolls?

Wave Serpent - energy field, weapons can have a max S8 and no more than 1d6 armour penetration against front and side armour.

So the numbers would be the same for the AV12 analysis.

Quin 242
26-04-2007, 15:13
ahh.. he didn't USE the melta rules.. I guess so.

Reinhard
26-04-2007, 16:42
I do my calculations in excel. As i can not put it here directly, I made some screenshots of the results. This should answer any questions about the probabilities. I did not check all calculations, so feel free to tell me any errors you find.

edit:
I have an error in the attached file.

the line for the fusion gun vs AV at more than 6" should be
1.73%, 10.84%, 27.10%, 33.87%, 21.17% and 5.29% for 0 to 5 penetrating hits.

Isambard
26-04-2007, 18:39
Lets do this properly then.
With a Wraithguard you have a 2/3 chance of hitting and a 2/3 chance of damaging, giving a 4/9 chance of doing damage vs a 5/9 chance of NOT doing damage (either missing or failing to glance/pen).

Given these odds and a bit of binomial probability you get the following odss:

Chance of 3/3 damage - 8.8%
Chance of 2/3 damage - 32.9%
Chance of 1/3 damage - 41.2%
Chance of 0/3 damage - 17.1%

Nice, eh.

At 12" with five fire dragons it is a bit different.....
The Dragons have a 2/3 chance of hitting and a 1/2 chance of damaging AV12 and a 1/3 chance of damaging AV13. This gives a damage probability of 1/3 vs AV12 and 2/9 chance of damaging AV13.

Taking this into account we get....

Chance of 5/5 damage - .4% (.05% for AV13)
Chance of 4/5 damage - 4.1% (.9% for AV13)
Chance of 3/5 damage - 16.5% (6.6% for AV13)
Chance of 2/5 damage - 32.9% (23.2% for AV13)
Chance of 1/5 damage - 32.9% (40.7% for AV13)
Chance of 0/5 damage - 13.2% (28.5% for AV13)

Looking at these numbers we get an expected value of 1 and 1/3 damaging hits for the Wraithguard vs 1 and 2/3 for the Dragins against AV12 and 1.11 against AV13. This doesn't give the whole story though.

Looking at the chance of getting 0 damage you can see the Dragons are a safer bet against AV12 but not AV13.

You have around a 7.5% chance of getting 3 or more damaging hits with the Fire Dragons against AV13 but an 8.8% chance of getting 3 damaging hits with the Wraithguard.

True, stats cant tell you everything but they can help you prepare.

marv335
26-04-2007, 18:43
something to bear in mind, the fusion guns used by the fire dragons are Ap1 so will penetrate rather than glance (SMF excepted)

mortiferum
26-04-2007, 21:33
Hey, thanks to ALL that have posted on this thread, I'm now fairly comfortable, reassured and understand how to calculate the following:

"Chance of atleast 1 Glancing / Penetrating hit"

Many of you have gone on and further calculated (which I thank you all for) the exact probabilities for 1, 2, 3, 4 and 5 hits glancing and penetrating.

Can you explain how this is done : How do I "sub catagorise" my "Chance of atleast 1 Glancing / Penetrating hit" into the individual probs for 1, 2, 3 ,4 or 5 glancing / penetrating hits (in the case of the Fire Dragons)?

Many thanks

Morty

BloodiedSword
26-04-2007, 21:46
For this you will need the Binomial theorem. Unfortunately, if you don't already know about this it's probably not worth trying to explain the theory behind it.

However, the good news is that you don't need to understand the theory in order to actually apply it.

So, let's say you want to work out the probability of A hits from B shots. You start by working out probability for an individual success and an individual failure. You then multiply

- (prob of individual success) to the power of number of successes you want, i.e. A
- (prob of individual failure) to the power of the number of failures, i.e. (B - A)
- the appropriate binomial coefficient, for which you can type B (the button that says "nCr" on your calculator) A.

The product of all these is the probability of A successes out of B trials.

mortiferum
26-04-2007, 21:57
- (prob of individual success) to the power of number of successes you want, i.e. A
- (prob of individual failure) to the power of the number of failures, i.e. (B - A)
- the appropriate binomial coefficient, for which you can type B (the button that says "nCr" on your calculator) A.

The product of all these is the probability of A successes out of B trials.

Thanks again for helping me out (wish I paid much more attention in my Maths classes when I was at school, no one told me it would come in useful for 40K gaming ;) )

I understood the first two bullet points but I don't get the third bullet point?
perhaps you could illustrate using an example, say 3 out of the 5 Fire Dragons getting a glance or penetrating hit?

Are you a Maths Prof in real life?

Sekhmet
26-04-2007, 22:55
Impossible to predict, given Dice dont follow the laws of probability, they follow the laws of physics and thermodynamics.

That said your 5 Fire Dragons have MORE SHOTS and thus increase their odds.

Dice do follow the laws of probability, it's just that you don't understand what probability is. That's all I'll say on this point.

Bloodied Sword got it all down, I'm lazy, so I won't do anything. :D

Just keep in mind that the expected number is usually far more useful (and easier to calculate!) than the exact probabilities of each result.

BloodiedSword
26-04-2007, 23:43
Well, seeing as I'm in a stats kind of mood I'll try and explain it :)

Let's label your Fire Dragons A, B, C, D and E.

Then if you just use the first two bullet points and try to work out the chance of getting 3 successes, what you actually work out is (for example) the chance that specifically A, B and C succeed while D and E fail.

But you don't care which 3 succeed as long as it's some combination of 3. It could be (A, D, E) or (B, C, D) or whatever for all you care.

That's where the nCr comes in, which stands for "number of ways of choosing r individuals from a group of size n".

EDIT: And no, I'm not a maths prof.. I'm a maths student actually :p

Rhamag
27-04-2007, 03:16
There is an easy to use Binomial Distribution funtion in MS Excel. You may need to install one of the Add-Ins first. Search in the Function pop-up. It's really easy to follow, and it'll give you the % for 1, 2, 3 etc sucesses.

wickedvoodoo
27-04-2007, 04:32
To the OP

Which models do you like better, WG or FD?

If the answer to which one is better develops into 2 pages of maths lessons then it is very likely that they are close enough that in most games it wont make a whole lot of difference. So pick the one you like the look of the most, easier that way.

Also gotta think about WG being more surviveable IMO and therefore more likely to do some damage after the intitial attack that the maths is working behind.

Ianos
27-04-2007, 08:38
Just think of it this way, FD's pack double the firepower per point yet are less survivable than wraithguard by half. In my book FD's are better because most of the time there will be nothing left of the enemy army to shoot them down or assault them. So if you want a unit that can move onto the enemy and deliver death unsuported but will not always guarantee results use WG, on the other hand if you want a specialist that will deliver massive firepower but must be supported to do it again go for FDs. Good luck in any case!

mortiferum
27-04-2007, 09:29
Well, seeing as I'm in a stats kind of mood I'll try and explain it :)

Let's label your Fire Dragons A, B, C, D and E.

Then if you just use the first two bullet points and try to work out the chance of getting 3 successes, what you actually work out is (for example) the chance that specifically A, B and C succeed while D and E fail.

But you don't care which 3 succeed as long as it's some combination of 3. It could be (A, D, E) or (B, C, D) or whatever for all you care.

That's where the nCr comes in, which stands for "number of ways of choosing r individuals from a group of size n".

EDIT: And no, I'm not a maths prof.. I'm a maths student actually :p

Once again many thanks for taking the time to explain, really appreciated - I tried doing the calculation by hand using the bullet points you outlined - it'll probablly come as no surprise I couldn't get it to work - really not sure how to work the nCr function on my calculator correctly :confused:

However, I did have much more succes using MS Excel and the function BINOMDIST - got all the answers I was looking for.

Once again, my thanks to all you Maths Profs out there :D

Incase anyone was wondering - based on the number crunching - I'll be using the Fire Dragons specifically to take out Tau Hammerheads and Eldar Falcons / Fire Prisms

Cheers

Morty

BloodiedSword
27-04-2007, 10:36
Who is to say that the calculations are just to decide which unit to take though?

Perhaps I'm in a situation where I have 2 units capable of firing on an AV 12 tank, but I need one of them to claim an objective or something. It would certainly help me if I knew things like which unit was most likely to destroy the target, which unit was most likely to cause at least one glancing/penetrating hit (to incapacitate the target from shooting next turn), etc.

Altashheth
27-04-2007, 20:10
Impossible to predict, given Dice dont follow the laws of probability, they follow the laws of physics and thermodynamics.

That said your 5 Fire Dragons have MORE SHOTS and thus increase their odds.

so random number generators dont follow the laws of probability??:wtf:

but they do follow the laws of thermodynamics, which utilise probability...

Are you an english lit major or equivalent?

susu.exp
28-04-2007, 00:58
Just keep in mind that the expected number is usually far more useful (and easier to calculate!) than the exact probabilities of each result.

No the expected value is overrated in mathhammer discussions. It would be a very useful value if these probability distributions were symetrical, but with GW games they tend not to be, but rather slanted to the left. So both the median and the mode will fall short of the mean, which means that both the most likely result of your shooting (mode) and the result you will roll equal to or worse than half the time (median) are lower than the expected number. Trusting E will lead you to curse your luck, because you will expect too much of your units most of the time.

Spotty
28-04-2007, 03:56
warhammer is supposed to be about having fun with some friends or with random strangers. I play eldar, and, quite honestly, I don't own single wraithuard model
because I don't like the way they look. Not because they are less effective at destroying stuff.(not sure if they are, and don't care) I play with fire dragons because i happen to think the models are bitchin. You are walking a thin line between Warhammer and Mathammer. Leave the dark side and embrace your future in 40k not 40%. Until then play with what you like and try not to become addicted to the false power of powergamers.

Good Luck young one,
Spotty:skull:

PH34R T3H M4TI-I4MM3R!!!!!!
PH34R \^/ITI-I MUCI-I PI-I34R!!!!!!!!

Rhamag
28-04-2007, 03:59
Thought I'd look at the shape of the curves that the binomial distribution of sucesses generates for a couple of different ork weapons, just for research purposes. The results have interested me, so if anyone who has stronger stat-fu can correct or guide my calculations, I'd be very grateful. Excel is my weapon of choice for this sort of thing.

I decided to start with 33 big shootas, giving a nice 99 dice. My target (please don't flame) was the industry-standard Tactical Marine. I generated a column of 1, 2, 3 etc. for the number of sucesses and used the handy BINOMDIST function to calculate the %.
The number of trials was 99 (Total Shotz)
The probability of sucess on each trial was 7.41% (1/3 * 2/3 * 1/3) (% Dead)

Then I added the Rokkits, converting via Total Points to get an equal "value" of rokkits. This gave:
Trials - 52
Prob Sucess - 27.78%

Graph (x-axis is Number Of Dead Marines)...
http://i67.photobucket.com/albums/h320/Rhamag/Warhammer%2040K/BigShoota-RokkitLauncha-Graph.jpg

Table...
http://i67.photobucket.com/albums/h320/Rhamag/Warhammer%2040K/BigShoota-RokkitLauncha-Table.jpg

So does this mean that, point-for-point, rokkits are better than big shootas at killing Marines? Or do I need to try again, but only use 3 of each, like in a real unit? Or have I screwed this up somewhere in the numbers?

Isambard
28-04-2007, 08:50
Why dont you use 8 rokkits and 5 big shootas, or multiples there of, as then the points balance out.

Did binomial distribution in school yesterday, its always a laugh.

If you are having problems with nCr on the calculator, you can always use Pascals Triangle, but it is a pain to draw.

susu.exp
28-04-2007, 10:43
So does this mean that, point-for-point, rokkits are better than big shootas at killing Marines? Or do I need to try again, but only use 3 of each, like in a real unit? Or have I screwed this up somewhere in the numbers?

The numbers are fine (though real units are a good excercize). But they are also misleading, because they assume that the marines get no cover safe. Even a cover safe of 6 has dramatic effects and the big shootas do better than the rokkits then. For a unit (13 Rapid firing Shootas + 3 BS versus 12 rapid firing Shootas + 3 Rokkits) it looks like this

A B C D
0 11.3% 9.6% 11.5% 13.7%
1 25.5% 24.5% 26.7% 28.7%
2 27.8% 28.9% 28.8% 28.4%
3 19.6% 21.0% 19.4% 17.8%
4 10.1% 10.6% 9.2% 7.9%
5 4.0% 4.0% 3.3% 2.6%
6 1.3% 1.1% 0.9% 0.7%
7 0.3% 0.3% 0.2% 0.1%
8 0.1% 0.1% 0.0% 0.0%
9 0.0% 0.0% 0.0% 0.0%
10 0.0% 0.0% 0.0% 0.0%

Where A is the 16 Orks with the BSs and B, C, D are the 15 Orks with the Rokkits with the marines in no, 6+ and 5+ cover respectively. With no cover the rokkits are a little more reliable, but even the slightest cover will make the BS the better choice. The points saved also buy another boy, making the unit more survivable and better in assaults. Against GEQ, the BS is even better compared to the Rokkit and the same goes for Termies. Your main anti-infantry upgrade is the BS, rokkits are for popping tanks.

elvinltl
29-04-2007, 11:52
Another thing to consider is whether they are independent or dependent events.
An easy way to figure these stuff out is to draw out a probability tree and link the sequence of events such as hitting and damage table.

The Binomial Theoream
P(X=r)= (n combination r)(p to the power of r)(q to the power of(n-r))
Let X be the random variable "the number of successes in the n trials of a binomial experiment"
p is the probability of success and q=1-p
r is the desired outcome

LirEdinSun
30-04-2007, 07:54
I recently used a squad of 5 Walocks on Jetbike led by a Farseer on Jet bike with Runes of Witnessing and Guide. all with Singing Spears.

IIRC Singing spears have strength 9 against Vehicles
Farseers have BS of 5 (2+ to hit) and Warlocks have BS 4 (3+ to hit)
Rune of Witnessing helps to make sure Guide always works.
Guide give you re-rolls.
Singing Spears means that even vehicles with Armour 14 glance on a 5+
So you can pretty much guarentee any vehicle will explode when attacked by this squad.

Also other benifites of the squad are 12" move and an additional 6" move during Assault phase to duck behind cover (annoyance factor).
Toughness =4, SV = 3+ (4+ invunerable)

Expensive but they work.