Fighting the Lizardmen - Leadership Rolls

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Fighting the Lizardmen - Leadership Rolls

Post by Zader »

I recently saw a discussion on probability calculation here, and initially was going to contribute to that; however since we'll be fighting the Lizardmen for SoC I thought it would be worth a seperate topic. Basically I asked myself, "Lizardmen test against Leadership rolls with 3d6 and take the best 2 instead of just 2d6 - from a probability standpoint, how much advantage is this really?"

Keep in mind I'm going to ignore BSB rerolls - at least for this discussion. ;) If you want to skip the math and go right to the analysis, look for the "So what does all this mean" section. ;)

So, the first part is relatively easy - calculating the 2d6 percentages. Basically this boils down to what is the probability that a certain leadership score will pass based on the unit's leadership. Basically given 2 dice, there are 36 possible combinations, and they add together for the following:

Code: Select all

Total Possible Rolls            Number Fraction Percentage
===== ====================      ====== ======== ==========
2     1,1                          1     (1/36)     2.78%
3     2,1:1,2                      2     (3/36)     8.33%
4     1,3:2,2:3,1                  3     (6/36)    16.67%
5     1,4:3,2:2,3:4,1              4     (10/36)   27.78%
6     1,5:2,4:3,3:4,2:1,5          5     (15/36)   41.67%
7     1,6:2,5:3,4:4,3:5,2:6,1      6     (21/36)   58.33%
8     2,6:3,5:4,4:5,3:6,2          5     (26/36)   72.22%
9     3,6:4,5:5,4:6,3              4     (30/36)   83.33%
10    4,6:5,5:6,4                  3     (33/36)   91.67%
11    5,6:6,5                      2     (35/36)   97.22%
12    6,6                          1     (36/36)    100%


This is straight forward - the possible rolls that total to a given number are listed, along with number of rolls equal or less than the number. (i.e. the probabilities are additive, so the probability of rolling an 8 or less, for example, is the probability of rolling a 2 + the probability of rolling a 3 + .. etc.)

Asleep yet? No? Ok, moving on.

So, the 3d6 chart is more complicated since you discard the highest die rolled, but the probabilities are calculated in the same way. There are 216 combinations (6*6*6). Here's the start of the chart - personally, I cheated and generated the numbers with a short perl script. ;)

Code: Select all

2 -  1,1,(1-6):1,(2-6),1:(2-6),1,1
3 -  1,2,(2-6):2,1,(2-6):1,(3-6),2:2,(2-6),1:(3-6),1,2:(3-6),2,1
4 -  1,3,(3-6):1,(4-6),3:(3-6),1,3:2,2,(2-6):2,(3-6),2:(3-6),2,2:3,1,(3-6):3,(3-6),1:(4-6),3,1
.
.
.
12 - 6,6,6


Numbers in parentheses denote it not mattering what the roll is as long as it's in the right range of numbers (like 4-6). So, for counting purposes, there are 16 possible combinations out of 216 that will result in exactly a 2 being rolled. The three dice being 1/1/1 , 1/1/2 , 1/1/3, etc. This results in the following numbers when you count them up:

Code: Select all

Total  Number  Fraction   Percentage
====== ======= ========== ==========
2         16    (16/216)  7.41%
3         27    (43/216)  19.91%
4         34    (77/216)  35.65%
5         36    (113/216) 52.31%
6         34    (147/216) 68.06%
7         27    (174/216) 80.56%
8         19    (193/216) 89.35%
9         12    (205/216) 94.91%
10        7     (212/216) 98.15%
11        3     (215/216) 99.54%
12        1     (216/216) 100%


Ok, so what does all this mean?

Basically this gives us a measure of how much better an equal leadership score really is for the lizardmen (i.e. Ld8 for Lizardmen means what if it were one of our units?) and also lets us estimate how easy or difficult it is to make unit types flee ...

So, lets compare - I found the numbers to be quite interesting. Lizardmen skinks with leadership 5 are going to run away, right? Well, compared to Harpies for example - not as often. Lizardmen leadership 5 is 52.31% chance to succeed, while Druchii Harpies at leadership 6 is a 41.67% chance to succeed. (Keep in mind I'm ignoring modifiers for right now). Skinks are closer to leadership 7 from a probability standpoint. Not quite as good a leadership as warriors though, but it's close. From a simplicity standpoint, you can more or less assume lizardman leadership is +1 to +2 what's listed as a rule of thumb estimate and be fairly close.

The middle to high end Lizardmen leadership is more of a difference in my mind since that's where more of the probability curve bunches up. Take a block of Saurus for example. They will succeed at a leadership check 89.35% of the time based on lizardman leadership of 8, whether supported by a general or not. We get around the same odds for a unit within 12" distance of a highborn general (91.67%), while our leadership 8 troops succeed checks 72% of the time. So keep this in mind - an unsupported block of lizardman infantry isn't necessarily going to be easier to break - and one in range of the general/BSB is going to be worse.

For a unit of Temple Guard (stubborn) with a Slann, they succeed leadership checks 95% of the time - regardless of casualties! The point is, no matter how many leadership checks you force on a TG unit w/ Slann, don't count on them breaking. (You can auto-break them with fear/terror and outnumber if they lose combat, but that doesn't require a test.) Either plan on killing this unit to a man (errrr...lizard), autobreak them through fear, or avoid them. I think that's important to know if you are fighting a Toad.

The other point is - Lizardmen units can be broken through combat, but it's harder to do than with the units of most armies. This is already an obvious point, but you can use the table to determine what level of success you are going to have - which means you can intelligently pick your fights since you have superior mobility.

example: 20 warriors vs 16 Saurus, warriors charge. Ok, from the chart, leadership 8 has a 89% chance to save - to get the odds in your favor you'll want < 50% chance of passing the test ... which means winning the combat by 4. Same number of ranks, lets assume musician and banner ... so you need to do 4 more wounds than your opponent. With 5 attacks, str 3 vs toughness 4 and 5+ save, it's not likely to happen. In fact, the warriors are likely going to lose the combat anyway unsupported.

Ok, lets use the same magic number (win combat by 4) with 20 warriors and a unit of 10 witch elves flanking - roughly the same points. The flank attack cancels ranks, so we're at +3 to start. Assuming the warriors take 3 wounds while dealing out 1 wound, the witch elves need to do at least 3 wounds more than what they suffer to win the combat by enough to have a good chance at forcing the Saurus to flee. (With 10-15 poisoned attacks, this is about even odds roughly that the Saurus will break and run - but still not definite).

Keep in mind this discussion is that of probabilities - not certainties. But with my dice rolls, I'd rather depend on stacking the odds in my favor rather than luck. ;)

For summary, essentially the main points are:

1) The lizardman leadership score is deceptive due to the 3d6 rolls to test. Use the above charts to get a feel for how hard a unit will be to break. Or, as a rule of thumb, you can assume they have a leadership +1 or +2 points of what's listed.
2) Temple Guard with a Slann aren't likely to miss a leadership test - period. You can autobreak them however with fear/terror causing units being involved in the combat. Plan appropriately.
3) Choose your fights wisely - determine whether or not you'll think you have the edge in combat before committing your troops to a fight whenever possible. Better mobility means it's possible to choose your fights, but it adds a lot of complexity to your movement phase - making it that much more important.
4) "Shooty" armies, assuming 3 turns of shooting, need to do a lot of damage - don't depend on Lizardmen running due to panic tests. Even an unscreened block of saurus (for example) have a 70.5% chance of passing all panic tests from shooting - even assuming you do enough casualties every turn to force a check. (If anyone is interested, I can estimate the number of shots it will take to have over a 50% of forcing a check every turn per Saurus block ... or it would be another good topic for discussion.)

Additional points from follow up comments:

From Reika:
Although it is implied by the numbers and whatnot, I just thought I'd mention something not specifically stated: expect any broken/panicked units to rally the next turn. As the numbers Zader kindly calculated for us show, even without a musician, pretty much the whole of the lizardman army has a better than average chance of rallying. Be sure movements/pursuits on your turn reflect this assumption.


Comments or constructive criticism are appreciated.
Last edited by Zader on Wed Apr 14, 2004 1:40 pm, edited 1 time in total.
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Vioxx
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Post by Vioxx »

<is really dizzy>

Way to take the game down to the base numbers, and suck all the fun out of playing. !eek!

Dont get me wrong, looks like you put a lot of work into this report-type-thing, but I like to look at the big picture. I understand the basics about odds and dice rolling, but to be honest, where is the fun in doing a ton of math just to figure out what the odds are for failing a leadership roll.

I dont trust odds, or statistics. The first thing you learn in that statistics dont prove anything. They just show trends. The fates are fickle, and Ive seen enough statistical anomalies (having rolled quite a few myself) to not trust all the math.

But, well done nonetheless...
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Post by Grogsnotpowwabomba »

I think this is a great resource for players to make good decisions on where to cause key Panic and Terror checks against the Lizardmen. This game is not all about math, but most of the best players make basic probabilities in their heads to determine the best decisions in a given circumstance. This chart just makes such decisions easier to accomplish with the added complexity that Lizardmen Cold Blooded LD creates.

It also proves, as I have said countless times, that Skinks are less reliable than LD7 humans.

Good work. I like it alot.
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Post by Vorchild »

Wow, I hadn't realized it was THAT disgusting.

Thanks for the analysis. As a rule of thumb, I don't do any such calculatins myself (there are enough of them in real life ;) ), but its always nice when someone comes out with such a nice presentation of the data. Hooray for lizardmen that all essentially get +2 to Ld! *sick*
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Post by Maraith tuerl »

Yeah, good presentation. The closest I came to that was to calculate using 6 tabs of a spreadsheet (to make a 6x6x6 matrix).... Try presenting *that* on a website... :roll:
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Post by Avatarofslaanesh »

one instance where it has been acceptable for people to figure out statistics like that, my Rugby coach had to take math thirty over agian to get into post secondary, but a2 months into the term he got accepted, he spent the entire term figuring out statistics to do with his blood angels

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Post by Drakken »

I think I have a headache now....

Nice work though. Definatly a point anybody whose played Lizzies alot knows fairly well though. Cold-Blooded effectivly makes Low LD not a weakness in any way.
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Post by Zader »

Drakken wrote:Definatly a point anybody whose played Lizzies alot knows fairly well though. Cold-Blooded effectivly makes Low LD not a weakness in any way.


Yup, I realize this isn't exactly news for folks experienced with playing lizardmen - but I thought it might help folks that haven't. I managed a massacre vs. a Lizardmen army over the weekend, but it was actually a really close game for awhile - it took some careful maneuvering to bring it off. Ask my CoK's what they think of the probability of getting hit with a spell for Irresistable Force and 12 hits on a 2d6 roll...which happened over the weekend. :shock:

And yes, I'm aware that all the numbers only show trends. Frankly putting the tables together wasn't any real work; 10 minutes to write the code to calculate the probabilities and then it was just the writeup. Basically I was bored at work waiting for a hardware vendor to finish tearing a drive array apart anyway. :roll:
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Post by Drakken »

numbers only show trends


Actually, I like to think of it more as the balancing Law of Averages. You can usually bank of average results when decidicing the sanity/intelligence of a manuver, as opposed to consider best/worst possible results. Whats really funnny is people with almost Anti-math backgrounds basing there thoughts on statistically results...
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Post by Zader »

Whats really funnny is people with almost Anti-math backgrounds basing there thoughts on statistically results...


Heh, it's a bit ironic - most modern electronics is based on quantum mechanics, which in turn is in essence pure probability theory and statistics. So if statistics aren't meaningful in any way then you can't be reading this ... since your computer can't possibly work. :lol:

I didn't realize that there were going to be negative responses just from a bit of math ... I thought I was being good. I didn't mention a collapse of the wavefunction even once. ;)
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Post by Darkadon »

Thanks for the probability tables :D :D :D . I was actually going to write a small C program to figure that stuff out but you beat me to it :evil: . I'm definitely going to print out your results and bring them with me to battle any time I play against the Lizards. It will serve dual purpose: I get a better idea of what I need to break a unit and I can play head games with my foe aswell.

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Post by Getsueimusha »

Oh wow... I JUST wanted this >.> like, a few hours before checking the forum. Though becuase I'm working on an all skink army of the southlands (whether or not I'll kill anything with it will be one thing, but it looks darn fun)
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Post by Zader »

Getsueimusha wrote:Oh wow... I JUST wanted this >.> like, a few hours before checking the forum. Though becuase I'm working on an all skink army of the southlands (whether or not I'll kill anything with it will be one thing, but it looks darn fun)


I've never played a southlands list, but I did play a Lizardmen army in 5th edition, which is pretty much the same thing. I have to admit though, it is tempting to pull one out since everyone considers me one of the local Druchii players - I think I'd get quite the reaction when I put 80 skinks on the table. ;)
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Post by Drakken »

So if statistics aren't meaningful in any way then you can't be reading this ... since your computer can't possibly work


:D I didn't say they weren't meaningful, I just said there are background that don't give a load about them. I didn't take a single statistics class my entire life, and had no basis of study on them at all outside of basic probibilities for war-gaming. Most Art (Fine or Regular) Majors don't have to touch the stuff, nor do a lot of specialist studies...
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Post by Zader »

:D I didn't say they weren't meaningful, I just said there are background that don't give a load about them. I didn't take a single statistics class my entire life, and had no basis of study on them at all outside of basic probibilities for war-gaming. Most Art (Fine or Regular) Majors don't have to touch the stuff, nor do a lot of specialist studies...


Whoops, wasn't referring to you in any way - one of the earlier posters actually. Sorry :)
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Post by Reika »

Although it is implied by the numbers and whatnot, I just thought I'd mention something not specifically stated: expect any broken/panicked units to rally the next turn. As the numbers Zader kindly calculated for us show, even without a musician, pretty much the whole of the lizardman army has a better than average chance of rallying. Be sure movements/pursuits on your turn reflect this assumption.
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Post by Zader »

Reika wrote:Although it is implied by the numbers and whatnot, I just thought I'd mention something not specifically stated: expect any broken/panicked units to rally the next turn. As the numbers Zader kindly calculated for us show, even without a musician, pretty much the whole of the lizardman army has a better than average chance of rallying. Be sure movements/pursuits on your turn reflect this assumption.


Excellent point. :) I'm going to add this to the original post in the summary section since it's a very important item. Thanks for bringing that up.
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Post by Veduhav »

I love math and statistics. Awesome post.
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Post by Shadowkiller venomblade »

i think it is ok, since my brother got lizzes a month ago....
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Post by Alkha'gthor »

A good piece of work Zader it would be interesting if we were to take the work a step further and calculate the probabilities for our scaly foes to pass break tests when charged by enemies that can auto-wound (WE) or troops that have a high enough strength to get through armour and wound easily (CoK and Koc, a mouthful there :) ). This would also aid us in destroying our enemies, if we do this as a problem with independent variables (I think been a while since I did in depth probability) we can work the answer through however this much maths may drain the fun completely out of the game.
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Post by Alkha'gthor »

Hmm got that wrong I think, I'll see if I can do it now.

Knights of Chaos (for SoC fans)
5 by 2 frontage charge a unit of Saurus with spears and a 5 by 5 formation. Saurus have +3 due to ranks and outnumber so the knights strike first neeeding threes to hit 6 attacks (non-chosen with champion) makes 4 hits and 3's to wound makes 2.67 wounds with no save. The steeds then strike 5 attacks needing 4's and 4's to hit and wound making 1.25 wounds with a 6+ save so .21 wounds saved making 1.04 caused for a total of 3.71 wounds the lizardmen strike back requiring 4's to hit and 4's to wound with 12.29 attacks making 6.145 hits and 3.0725 wounds with a 3+ armour save for the knights making 2.058575 saves for a total of 1.013925 wounds therefore the knights have won combat by 2.7 meaning they have not overcome the Saurus but a charge by 8 DE in the flank as stated cancels this +3 and the attacks made by 4 of these require 3's and 5's making 0.8844 wounds 0.292 of which shall be saved the 3 saurus that can strike back have 6 attacks needing 4's and 3's the DE take 2.01 wounds and save .336 resulting in 1.674 wounds so the DE's now win combat by 1.03 on the original but with the cancelling of the lizards ranks the results swings to 3.03 making the lizards pass a break test on 5 which with three dice have a 52.31% to pass (thanks Zadar) so it can be seen that the Druchii will break the Saurus almost half the time and so it shows that even a combined arms charge by elite troops and average troops will struggle to break the enemy and so it shows that the DE need to rely on movement as mentioned and must always pursue. Sorry to drag on but this just confirms the results of Zadar and shows that in the game the stats while not definitive show that the struggle will be a hard one.
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Post by Zader »

Alkha'gthor wrote:A good piece of work Zader it would be interesting if we were to take the work a step further and calculate the probabilities for our scaly foes to pass break tests when charged by enemies that can auto-wound (WE) or troops that have a high enough strength to get through armour and wound easily (CoK and Koc, a mouthful there :) ). This would also aid us in destroying our enemies, if we do this as a problem with independent variables (I think been a while since I did in depth probability) we can work the answer through however this much maths may drain the fun completely out of the game.


Edit - mistakenly had the CoK doing wounds on 2+ and not 3+ -> corrected this and added some data from later in the discussion. (Namely the combined Knight/Cold One probability table, and the Saurus attacks back for the case of wounds = 3)

Well, I'm not sure it's worth it to take it to the extreme since we're not talking about enough models to be statistically significant, but we can at least look at some numbers to make some rule of thumb estimates.

Take the example of the CoK charging a saurus block for example. It's a straightforward calculation of the probability of causing a wound:

Hit on 3+, wound on 3+, and modified armor save of 6+ due to the strength of the hit becomes:

Knight wound die rolls = (3-6)(3-6)(1-5) = 4/6*5/6*5/6 = 37.0%

And for the cold ones (last roll changes due to the str 4 hit)
Cold One wound die rolls = (4-6)(4-6)(1-4) = 3/6*3/6*4/6 = 16.67%

So if we're looking for winning the combat by 4 to have a > 50% chance of the unit breaking, lets look at the odds. We'll assume flank attack (+1, +1 ranks), but outnumbered (-1).

This gets a little complicated, but lets ignore trying to come up with a nice neat formula that's probably going to be hard to understand and just calculate things the long way. We'll cheat a bit however. ;) There are basically two possibilities, that either a wound will occur or a wound won't occur. Simple enough. Lets express this by saying a 0 is a failure to wound, and a 1 equals a wound and look at all the possibilities for 5 knights attacking - ignoring the cold ones which will be handled the same way. We have:

00000 - all knights fail to wound
00001 - One knight on the right side of the formation wounds
00010 - One knight to the just to the far right wounds
.
.
.
11111 - all knights wound

And yes, I am aware we don't really care about which knight wounds, just if they do so or not - but since we're going to do the calculations with a small perl script, we don't really care. :) It will aggregate the combinations for us based on the number of wounds.

Basically we get these results - I'll include the script later:

0 wounds -> 9.90%
1 wounds -> 29.10%
2 wounds -> 34.24%
3 wounds -> 20.14%
4 wounds -> 5.92%
5 wounds -> 0.70%

So the most probable result, is to do 1-3 wounds. Now lets look at the cold ones:

Probability of 0 wounds: 40.11%
Probability of 1 wounds: 40.20%
Probability of 2 wounds: 16.12%
Probability of 3 wounds: 3.23%
Probability of 4 wounds: 0.32%
Probability of 5 wounds: 0.01%

So basically we can expect either 0 or 1 wounds from the Cold Ones.

So looking at the numbers, and without a champion, noble, or magical banner in the unit, we can reasonably expect 1-4 wounds from the unit - not counting any casualties that we take back. See the followup on page 2 for the calculation of attacks back for the combined knight/cold ones wounds = 3 case, which makes our final table:

0 : 3.9709
1 : 15.6518
2 : 27.0277
3 : 18.65
4 : 16.9619
5 : 7.1072
6 : 1.9987
7 : 0.3719
8 : 0.0436
9 : 0.0028
10 : 0.0001

So our chances of doing 3+ wounds are: 45.13% (Note, in the previous edit I had the Knights wounding on 2+ when it should have been 3+ - this changed the numbers. Oops. :oops: )

Our chances of doing the 3 wounds or more are about even for an unsupported unit of CoK's.

So, all this math basically tells us what we already know. ;) An attack unsupported by magic banners/nobles/etc (or other units) from a group of CoK's to the flank of a Saurus block will likely result in a first round combat loss to the lizards, but the chances are about even that we'll do enough wounds back to cause a realistic chance to break the unit - with things getting uglier in later rounds when the CoK's str drops back to 3. An attack to the front is likely to result in a combat tie or loss, with the ranks/outnumber bonus to the Saurus at +3 (assuming there is an additional rank of CoK's).

This does however show the value of the hydra banner (increasing the attacks, thus the probability of getting 4+ CR result) or the banner of slaughter (+d3 CR). Having a noble and/or champion will help as well.

Here's the perl script I used for the calculations for anyone interested - basically it just uses binary (0/1) for whether it's a hit or miss, and runs through all the possible results, adding the probabilities to the number of wounds. (And yes, I'm aware that the numbers don't exactly add up to 100% - it's rounding error due to not using the exact percentages)

Code: Select all

#!/usr/bin/perl

$wound_chance = .370;
$fail_chance = 1 - $wound_chance;

# Create distribution table

for $i (0 .. 31) {
        $prob = 1;
        $count = 0;
        for $j (0 .. 4) {
                if ($i & (1 << $j)) {
                        $count++;
                        $prob *= $wound_chance;
                } else {
                        $prob *= $fail_chance;
                }
        }
        $hits{$count}+= $prob;
}

foreach $k (sort keys %hits) {
        $percent = $hits{$k} * 100;
        printf "Probability of $k wounds: %5.2f\%\n",$percent;
}
Last edited by Zader on Sat Apr 17, 2004 12:25 pm, edited 1 time in total.
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
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Post by Zader »

Knights of Chaos (for SoC fans)
5 by 2 frontage charge a unit of Saurus with spears and a 5 by 5 formation.
.
.


Yup, when I'm doing the estimates in my head, I just use the averages and round up or down as appropriate. The only real reason I did it the long way for the CoK example was to show the actual distribution of ppercentages. Knowing that the CoK will on average do 2 wounds, but I thought it might be interesting to see what the probability was of meeting our goal of 4 wounds on an unsupported flank charge. It's interesting to note while the average is 2.315 wounds, it's equally likely (roughly) that there will be either 2 or 3 wounds.
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Alkha'gthor
Corsair
Posts: 87
Joined: Mon Mar 17, 2003 1:56 am

Post by Alkha'gthor »

Yup, when I'm doing the estimates in my head, I just use the averages and round up or down as appropriate. The only real reason I did it the long way for the CoK example was to show the actual distribution of ppercentages. Knowing that the CoK will on average do 2 wounds, but I thought it might be interesting to see what the probability was of meeting our goal of 4 wounds on an unsupported flank charge. It's interesting to note while the average is 2.315 wounds, it's equally likely (roughly) that there will be either 2 or 3 wounds.


Yeah it is funny that the probability is higher than the average roughly maths makes fools of us all, and all of this proves categorically that the lizards in particular the saurus must be combined charged or hit in the flanks by elite regiments for the Druchii to win by enough this been 4 to enforce the break test with a 50% or greater chance of breaking.
Look into the eyes of death and know that your life is meaningless, in my grand scheme you are less than dirt. Alkha'gthor, Lord of Torment
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Malachi
Malekith's Best Friend
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Joined: Wed Mar 12, 2003 9:27 pm

Post by Malachi »

@ Zader this is really interesting stuff. It is still out there for my comprehension but I think it is very useful. More of this work nailed out the long and hard way with recommendations like you have done would be very useful for all generals.

I think the shooting analysis would be very interesting. Please start peril scripting!
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