Fighting the Lizardmen - Leadership Rolls

Where the great threads of Druchii.net are kept

Moderator: The Dread Knights

User avatar
Maraith tuerl
Malekith's Best Friend
Posts: 1250
Joined: Fri May 23, 2003 12:50 pm

Post by Maraith tuerl »

@Zader> One quick point/question. It appears that your calculations kick out the chances of doing exactly X wounds. Really, what we're interested in, is doing >X wounds, no? I mean, we're interested in the 'magic number' that will let us autobreak, or else win with enough to likely break.

So, in your example, we have a 10 strong CoK regiment going against a 25 strong Saurus regiment. For the autobreak, you need 6 wounds. Not all that likely, w/o the Hydra banner (intuitively) (I get it somewhere around 8-9%, based on your #s).

So, let's do your example, flank charge by the CoK. You start up by one (rank vs. no ranks, banners cancel, flank cancels outnumber). You need to win by 4 to have a reasonable chance of breaking the unit, imo (that gives them a -4 Ld penalty, which brings them down to 35% chance to pass, if I'm reading your table right). So, we're looking for 3 wounds (and with 3 wounds, it's reasonable to assume the remaining 2 Sauri won't get a wound on you; figure it's roughly 2 hits, slightly better than 1 wound, but a 3+ save)

Now, the thing is, you don't need 3 wounds. You need 3 *or more* wounds. So, really, you're looking at something like:

p(Knights get 3+ wounds solo) + p(Knights get 2 wounds * CO get 1+ wounds) + p(Knights get 1 wound * CO get 2+ wounds) + p(Knights get 0 wounds * CO get 3+ wounds).

I come up w/ 69.23%, based on that formula, of getting 3+ wounds in any combination.
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

Malachi wrote:@ 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!


lol, ok, I'll take a look at some shooting examples this weekend. Any particular scenerio you want me to model? I guess I could abuse that poor block of Saurus again.

On another note though, what can I do to make things more understandable? I'm trying to avoid as many of the math technical terms as possible to make things interesting - part of why the writeups have been so long. I could express things as combinations, permutations, and whatnot but I thought doing things the "long way" would be a bit easier to understand the theory. Any suggestions?
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

Edit - I reversed the ranks for the Saurus. I meant to say 4 ranks of 5, not 5 ranks of 4.

Maraith Tuerl wrote:@Zader> One quick point/question. It appears that your calculations kick out the chances of doing exactly X wounds. Really, what we're interested in, is doing >X wounds, no? I mean, we're interested in the 'magic number' that will let us autobreak, or else win with enough to likely break.

So, in your example, we have a 10 strong CoK regiment going against a 25 strong Saurus regiment. For the autobreak, you need 6 wounds. Not all that likely, w/o the Hydra banner (intuitively) (I get it somewhere around 8-9%, based on your #s).

So, let's do your example, flank charge by the CoK. You start up by one (rank vs. no ranks, banners cancel, flank cancels outnumber). You need to win by 4 to have a reasonable chance of breaking the unit, imo (that gives them a -4 Ld penalty, which brings them down to 35% chance to pass, if I'm reading your table right). So, we're looking for 3 wounds (and with 3 wounds, it's reasonable to assume the remaining 2 Sauri won't get a wound on you; figure it's roughly 2 hits, slightly better than 1 wound, but a 3+ save)

Now, the thing is, you don't need 3 wounds. You need 3 *or more* wounds. So, really, you're looking at something like:

p(Knights get 3+ wounds solo) + p(Knights get 2 wounds * CO get 1+ wounds) + p(Knights get 1 wound * CO get 2+ wounds) + p(Knights get 0 wounds * CO get 3+ wounds).

I come up w/ 69.23%, based on that formula, of getting 3+ wounds in any combination.


Well, you made the main point - I screwed up the number of wounds needed for starters since I forgot the +1 rank for the CoK. :) I meant to model it against 20 Saurus though, 4 ranks of 5. My mistake was that at the start I had assumed 10 Knights, but then forgot and did the analysis based on 5 knights - my error. So that's one part.

And yes, I get the roughly the same number you do based on 3 wounds. Basically I was looking at > 50% probabilities as a "decision test" on 4 wounds, and if 4 wounds are required that brings the probability to < 50%. (38.8% from my calculations). I skipped right to the conclusion since for 4w it was under 50% - but my base assumption was wrong, as you pointed out. :)

Since only 3 wounds are required, that brings the probability up to 66.91% - which includes any amount of wounds > 3.

Here's the probability table I'm using to get the number based on 0-10 wounds (numbers are percentages):

0 : 1.7929
1 : 9.5181
2 : 21.7756
3 : 28.0734
4 : 22.4427
5 : 11.5630
6 : 3.8781
7 : 0.8368
8 : 0.1111
9 : 0.0080
10 : 0.0002

p(w >= 3) = 100 - p(w < 3) = 66.91%

This is based on the following if we're doing it the long way - although I'm going to use a shorthand notation. Read this as p(0 knight wounds)*p(0 cold one wounds), etc.:

0 - 0k0c
1 - 1k0c + 0k1c
2 - 2k0c + 1k1c + 0k2c
3 - 3k0c + 2k1c + 1k2c + 0k3c
4 - 4k0c + 3k1c + 2k2c + 1k3c + 0k4c
5 - 5k0c + 4k1c + 3k2c + 2k3c + 1k4c + 0k5c
6 - 5k1c + 4k2c + 3k3c + 2k4c + 1k5c
7 - 5k2c + 4k3c + 3k4c + 2k5c
8 - 5k3c + 4k4c + 3k5c
9 - 5k4c + 4k5c
10 - 5k5c

This look ok now? Oh, one very interesting thing this brings up - it looks like 5 knights fails the > 50% chance while the added rank for 9-10 knights will probably succeed. Interesting stuff!
Last edited by Zader on Sat Apr 17, 2004 12:12 pm, edited 2 times in total.
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

whoops...

Post by Zader »

Hmm...ok, one last item and we can put the example to sleep - namely the attacks back. In the case of doing 3 wounds we have to take an attack back into account ... so here goes.

Lets just look at modifying wounds > 3 since it's the simple case; if we don't win by 3 we won't hit our number of 50% chance to break anyway and we only have to deal with a single attacker. (I suppose we really should look at the odds of losing the combat entirely). So we can still ignore attacks back for any 4+ wounds cases anyway since there won't be any attacks back:

So basically we just need to modify the chances of doing exactly 3 wounds that I posted earlier -> with 4+ there are no attacks back.

Our calculation is just the lizardmen striking back with 2 attacks doing no wounds p(L0), multiplied by the probability of doing 3 wounds in any combination. p(w3). So p(L0)*p(w3) = 19.49%

For the 3+ CR result we now have:

Edit: See the revised table on the original CoK discussion - original calculations were off.

Which makes the probability just under 50%. Basically a flip of the coin, which keeps the game exciting right? ;)

Here's the modified wounds = 3w case "proof" for critique.

Code: Select all

p(new_w3) =
      p(k3)p(c0)p(L0) + p(k2)p(c1)(l0) + p(k1)p(c2)(l0) + p(k0)p(c3)p(l0) +
      p(k4)p(c0)p(L1) + p(k3)p(c1)(L1) + p(k2)p(c2)(l1) + p(k1)p(c3)p(l1) + p(k0)p(c4)p(l1) +
      p(k5)p(c0)p(l2) + p(k4)p(c1)(l2) + p(k3)p(c2)(l2) + p(k2)p(c3)p(l2) + p(k1)p(c4)(l2) + p(k0)p(c5)p(l2)

Simplifying to:

      p(l0) * (p(k3)p(c0) + p(k2)p(c1) + p(k1)p(c2) + p(k0)p(c3)) +
      p(l1) * (p(k4)p(c0) + p(k3)p(c1) + p(k2)p(c2) + p(k1)p(c3) + p(k0)p(c4)) +
      p(l2) * (p(k5)p(c0) + p(k4)p(c1) + p(k3)p(c2) + p(k2)p(c3) + p(k1)p(c4) + p(k0)p(c5))

Further simplified to:

      p(l0) * (p(3w)) +
      p(l1) * (p(4w)) +
      p(l2) * (p(5w))

But ... if there were 4w+ between the cold ones and knights, we can ignore p(l1) p(l2) entirely since there were no attacks back possible!  This makes it:

p(l0) * p(3w)

This makes the calculation easy since we already have the probabilities for the number of wounds done from before.


So the Saurus probability table goes:

0 - 69.44%
1 - 27.78%
2 - 2.778%

Questions? Comments?
Last edited by Zader on Sat Apr 17, 2004 12:29 pm, edited 1 time in total.
Prince of shadow
Noble
Posts: 410
Joined: Wed Dec 17, 2003 2:45 pm
Location: Naggaroth

Post by Prince of shadow »

Why dont you try this...

Cold one knight regiment
9 cold one knights with champion
Warbanner

1 cr from rank
2 cr from banner
1 maybe from outnumber

Adds 3-4 combat resolution

BsB join regiment with hydra banner

10 attacks from cold one kights
6 hit
s5 to t4; 3 wounds

10 attacks from cold ones
5 hit
s4 to t4; 2 wounds

3 attacks from noble
2 hit
s4 to t4
1 wound

total generated cr is
6 cr + 4cr =10 cr
10cr - 4cr=6cr

From this we can see that the saurus regiment would lose the combat by 6cr.Enough to autobreak.
Man who fought bravely die quickly, while man who cower and run live to fight another day
User avatar
Alkha'gthor
Corsair
Posts: 87
Joined: Mon Mar 17, 2003 1:56 am

Post by Alkha'gthor »

Just to add guys if we were to hit the saurus in the flanks with our ten knights and the saurus took the three wounds (most likely) this means that the knights face the 4 attacks for 2 hits and need 3's to wound making 1.33 wounds and the knights need to make a 3+ save so save 2/3 of the 4/3 caused hence .89 saved so 1.33 - 0.89 = 0.44 so the probability been the wound saved meaning our cold one knights now outnumber as well on average by 3, US 20 to US 17. So 3 wounds to 0, +1 rank, +1 ountumber, +1 flank attack making the most probable victory margin +6. So even if our COK didn't cause fear the Saurus would only have 7.41% to pass a break test. So our conclusion must lie that heavy cavalry with a reasonable no. of attacks can beat the saurus into pulp provided they hit a flank and can outnumber even with the lizards 3d6 leadership tests.
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
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

I have a new program that also handles attacks back and shows the final CR probabilities for front, side, and rear attacks. I'll post the results for the CoK vs Saurus this evening. (I'm at work so I can't do it now, and it contains the proprietary GW data, so I don't think I can post the actual program). After that I'll post a final analysis and summary of the data and we can pretty much declare the topic dead - after all, this started as a probability discussion of lizardman leadership and then wandered far afield.

Incidently I wrote the new program to be a general case probability calculator for combats, which means I can do the probability model for any 1v1 combats by just changing the attacker and defender. I need to do a final revision to more easily handle multiple attackers - but even the initial results are rather interesting. I'll post some of the results this evening as new topics for discussion.

Now if I could just tie this into the army builder databases, I could do the probability models for any units fighting without having to input them by hand. Hmmmm.... now tie this into a java front end for movement, and we're 3/4's the way to having online warhammer. I know GW wouldn't allow this, but it would be an interesting way to quickly test army lists and battle tactics without having to fight a full blown battle for a standalone version ...
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

Ok, I have to prep my CoS army for a 2000 pt game tomorrow after work vs. a Chaos Khorne army, so without further ado...

Special thanks go to Maraith Tuerl for pointing out my initial calculation mistake and prompting me to review the numbers, Alkha'gthor for taking an alternate approach to the numbers which I think validate my final results, and everyone else that was interested that pushed me to just write the program so I'd quit making simple mistakes. :)

Here's the 3 probability tables, for flank, rear, and frontal attacks of a 5x2 group of Cold One Knights vs. a 5x4 block of Saurus. Items of interest:

1) The flank attack kicks some serious lizard butt.
2) The rear attack is nearly as good, but the Saurus have just a slightly better chance due to the extra model - even with the additional +1 to CR.
3) The charge to the front will probably autobreak the Saurus since at the end of the combat we will likely outnumber and cause fear. However, if the Lizardman has the spawning of Tlazcotl (immune to psychology) be advised that our CR probably isn't high enough to make the Saurus run - so we'll get bogged down in combat.

Even if the Saurus have the Tlazcotl spawning, our +CR is going to be high enough to force a break test with a good probability of success.

Here's the 3 tables - the last very long block is in case someone wants to validate the results. All the calculations are included. Note that CR results that were < 1% probability are not displayed just for brevity.

Flank:

Code: Select all

=====================================================
...Attack is to the flank, +1 to CR.  (1)
...Attacker has standard, +1 to CR. (2)
...Defender has standard, -1 to CR. (1)
...Attacker has 2 Ranks, 1 modifier to CR. (2)
...Defender attacked to the rear or flank, no rank bonus.
...see table below for outnumber bonus
=====================================================
Combat Results table (includes CR modifiers)
=====================================================
0 CR result -> (...Defender Outnumbers, -1 to CR)  3.49%
2 CR result -> (US is tied, no outnumber bonus)  8.93%
4 CR result -> (...Attacker Outnumbers, +1 to CR) 16.51%
5 CR result -> (...Attacker Outnumbers, +1 to CR) 22.19%
6 CR result -> (...Attacker Outnumbers, +1 to CR) 21.22%
7 CR result -> (...Attacker Outnumbers, +1 to CR) 16.95%
8 CR result -> (...Attacker Outnumbers, +1 to CR)  7.09%
9 CR result -> (...Attacker Outnumbers, +1 to CR)  1.99%
=====================================================
Chance of achieving CR of 4 or more: 86.36%
=====================================================


Rear:

Code: Select all

=====================================================
...Attack is to the rear, +2 to CR.  (2)
...Attacker has standard, +1 to CR. (3)
...Defender has standard, -1 to CR. (2)
...Attacker has 2 Ranks, 1 modifier to CR. (3)
...Defender attacked to the rear or flank, no rank bonus.
...see table below for outnumber bonus
=====================================================
Combat Results table (includes CR modifiers)
=====================================================
0 CR result -> (...Defender Outnumbers, -1 to CR)  1.58%
1 CR result -> (...Defender Outnumbers, -1 to CR)  4.73%
3 CR result -> (US is tied, no outnumber bonus) 10.59%
5 CR result -> (...Attacker Outnumbers, +1 to CR) 17.69%
6 CR result -> (...Attacker Outnumbers, +1 to CR) 21.93%
7 CR result -> (...Attacker Outnumbers, +1 to CR) 20.11%
8 CR result -> (...Attacker Outnumbers, +1 to CR) 13.39%
9 CR result -> (...Attacker Outnumbers, +1 to CR)  7.09%
10 CR result -> (...Attacker Outnumbers, +1 to CR)  1.99%
=====================================================
Chance of achieving CR of 4 or more: 82.62%
=====================================================


Front:

Code: Select all

=====================================================
...Attacker has standard, +1 to CR. (1)
...Defender has standard, -1 to CR. (0)
...Attacker has 2 Ranks, 1 modifier to CR. (1)
...Defender has 4 Ranks, 3 modifier to CR. (-2)
...see table below for outnumber bonus
=====================================================
Combat Results table (includes CR modifiers)
=====================================================
-5 CR result -> (...Defender Outnumbers, -1 to CR)  1.58%
-4 CR result -> (...Defender Outnumbers, -1 to CR)  4.73%
-2 CR result -> (US is tied, no outnumber bonus) 10.59%
0 CR result -> (...Attacker Outnumbers, +1 to CR) 17.69%
1 CR result -> (...Attacker Outnumbers, +1 to CR) 21.93%
2 CR result -> (...Attacker Outnumbers, +1 to CR) 20.11%
3 CR result -> (...Attacker Outnumbers, +1 to CR) 13.39%
4 CR result -> (...Attacker Outnumbers, +1 to CR)  7.09%
5 CR result -> (...Attacker Outnumbers, +1 to CR)  1.99%
=====================================================
Chance of achieving CR of 4 or more:  9.51%
=====================================================


Entire table for flank attack

Code: Select all

==============================================================
...Cold One Knight needs 3 To hit
...Cold One Knight needs 3 To wound
 ... Attacker strength 5 has an armor save modification of -2
Defender has a modified armor save of 6...
0.6667 * 0.6667 * 0.8333 * 1.0000 = 0.3704
==============================================================
...Saurus needs 4 To hit
...Saurus needs 3 To wound
 ... Attacker strength 4 has an armor save modification of -1
Defender has a modified armor save of 3...
0.5000 * 0.6667 * 0.3333 * 1.0000 = 0.1111
==============================================================
...Cold One needs 4 To hit
...Cold One needs 4 To wound
 ... Attacker strength 4 has an armor save modification of -1
Defender has a modified armor save of 5...
0.5000 * 0.5000 * 0.6667 * 1.0000 = 0.1667
==============================================================
Cold One Knight chance to wound --->  0.370
==============================================================
Cold One chance to wound --->  0.167
==============================================================
Saurus chance to wound --->  0.111
==============================================================
...calculating for flank attack.
==============================================================
==============================================================
Individual to-hit probabilities Cold One Knight
==============================================================
Hits: 0 --> 0.6296
Hits: 1 --> 0.3704
==============================================================
==============================================================
Individual to-hit probabilities Saurus
==============================================================
Hits: 0 --> 0.7901
Hits: 1 --> 0.1975
Hits: 2 --> 0.0123
==============================================================
==============================================================
Individual to-hit probabilities Cold One
==============================================================
Hits: 0 --> 0.8333
Hits: 1 --> 0.1667
==============================================================
=====================================================
Preliminary Probability tables -> Cold One Knight
=====================================================
0 wounds ->  9.90%
1 wounds -> 29.10%
2 wounds -> 34.24%
3 wounds -> 20.14%
4 wounds ->  5.92%
5 wounds ->  0.70%
==============================================================
=====================================================
Preliminary Probability tables -> Cold One
=====================================================
0 wounds -> 40.19%
1 wounds -> 40.19%
2 wounds -> 16.08%
3 wounds ->  3.22%
4 wounds ->  0.32%
5 wounds ->  0.01%
==============================================================
=====================================================
Preliminary Probability tables -> Saurus
=====================================================
0 wounds -> 38.97%
1 wounds -> 38.97%
2 wounds -> 17.05%
3 wounds ->  4.26%
4 wounds ->  0.67%
5 wounds ->  0.07%
6 wounds ->  0.00%
7 wounds ->  0.00%
8 wounds ->  0.00%
==============================================================
...Combined total chances to wound: 10
=====================================================
Combined Probability tables -> Cold One Knight, Cold One
==============================================================
0 wounds ->  3.98%
1 wounds -> 15.67%
2 wounds -> 27.05%
3 wounds -> 26.85%
4 wounds -> 16.95%
5 wounds ->  7.09%
6 wounds ->  1.99%
7 wounds ->  0.37%
8 wounds ->  0.04%
9 wounds ->  0.00%
10 wounds ->  0.00%
==============================================================
Defender limited to maximum of 4 attacks back...
Attacker can do 0 to 10 wounds...
     ...0 wounds received -> Defender gets 4 attackers striking back
==============================================================
Individual to-hit probabilities Saurus
==============================================================
Hits: 0 --> 0.7901
Hits: 1 --> 0.1975
Hits: 2 --> 0.0123
==============================================================
=====================================================
     (Modified defender probability table)
=====================================================
     8 attacks doing 0 wounds: 0.389744
     8 attacks doing 1 wounds: 0.389744
     8 attacks doing 2 wounds: 0.170513
     8 attacks doing 3 wounds: 0.042628
     8 attacks doing 4 wounds: 0.006661
     8 attacks doing 5 wounds: 0.000666
     8 attacks doing 6 wounds: 0.000042
     8 attacks doing 7 wounds: 0.000001
     8 attacks doing 8 wounds: 0.000000
=====================================================
     case a(0), d(0) (0) : 0.015499
     case a(0), d(1) (-1) : 0.015499
     case a(0), d(2) (-2) : 0.006781
     case a(0), d(3) (-3) : 0.001695
     case a(0), d(4) (-4) : 0.000265
     case a(0), d(5) (-5) : 0.000026
     case a(0), d(6) (-6) : 0.000002
     case a(0), d(7) (-7) : 0.000000
     case a(0), d(8) (-8) : 0.000000
     ...1 wounds received -> Defender gets 3 attackers striking back
==============================================================
Individual to-hit probabilities Saurus
==============================================================
Hits: 0 --> 0.7901
Hits: 1 --> 0.1975
Hits: 2 --> 0.0123
==============================================================
=====================================================
     (Modified defender probability table)
=====================================================
     6 attacks doing 0 wounds: 0.493270
     6 attacks doing 1 wounds: 0.369953
     6 attacks doing 2 wounds: 0.115610
     6 attacks doing 3 wounds: 0.019268
     6 attacks doing 4 wounds: 0.001806
     6 attacks doing 5 wounds: 0.000090
     6 attacks doing 6 wounds: 0.000002
=====================================================
     case a(1), d(0) (1) : 0.077309
     case a(1), d(1) (0) : 0.057982
     case a(1), d(2) (-1) : 0.018119
     case a(1), d(3) (-2) : 0.003020
     case a(1), d(4) (-3) : 0.000283
     case a(1), d(5) (-4) : 0.000014
     case a(1), d(6) (-5) : 0.000000
     ...2 wounds received -> Defender gets 2 attackers striking back
==============================================================
Individual to-hit probabilities Saurus
==============================================================
Hits: 0 --> 0.7901
Hits: 1 --> 0.1975
Hits: 2 --> 0.0123
==============================================================
=====================================================
     (Modified defender probability table)
=====================================================
     4 attacks doing 0 wounds: 0.624295
     4 attacks doing 1 wounds: 0.312148
     4 attacks doing 2 wounds: 0.058528
     4 attacks doing 3 wounds: 0.004877
     4 attacks doing 4 wounds: 0.000152
=====================================================
     case a(2), d(0) (2) : 0.168852
     case a(2), d(1) (1) : 0.084426
     case a(2), d(2) (0) : 0.015830
     case a(2), d(3) (-1) : 0.001319
     case a(2), d(4) (-2) : 0.000041
     ...3 wounds received -> Defender gets 1 attackers striking back
==============================================================
Individual to-hit probabilities Saurus
==============================================================
Hits: 0 --> 0.7901
Hits: 1 --> 0.1975
Hits: 2 --> 0.0123
==============================================================
=====================================================
     (Modified defender probability table)
=====================================================
     2 attacks doing 0 wounds: 0.790123
     2 attacks doing 1 wounds: 0.197531
     2 attacks doing 2 wounds: 0.012346
=====================================================
     case a(3), d(0) (3) : 0.212155
     case a(3), d(1) (2) : 0.053039
     case a(3), d(2) (1) : 0.003315
case a(4), d(none) (4): 0.169464
case a(5), d(none) (5): 0.070940
case a(6), d(none) (6): 0.019937
case a(7), d(none) (7): 0.003716
case a(8), d(none) (8): 0.000440
case a(9), d(none) (9): 0.000030
case a(10), d(none) (10): 0.000001
=====================================================
Combat Results table (wounds only)
=====================================================
-8 wounds ->  0.00%
-7 wounds ->  0.00%
-6 wounds ->  0.00%
-5 wounds ->  0.00%
-4 wounds ->  0.03%
-3 wounds ->  0.20%
-2 wounds ->  0.98%
-1 wounds ->  3.49%
0 wounds ->  8.93%
1 wounds -> 16.51%
2 wounds -> 22.19%
3 wounds -> 21.22%
4 wounds -> 16.95%
5 wounds ->  7.09%
6 wounds ->  1.99%
7 wounds ->  0.37%
8 wounds ->  0.04%
9 wounds ->  0.00%
10 wounds ->  0.00%
=====================================================
...Attack is to the flank, +1 to CR.  (1)
...Attacker has standard, +1 to CR. (2)
...Defender has standard, -1 to CR. (1)
...Attacker has 2 Ranks, 1 modifier to CR. (2)
...Defender attacked to the rear or flank, no rank bonus.
...see table below for outnumber bonus
=====================================================
Combat Results table (includes CR modifiers)
=====================================================
0 CR result -> (...Defender Outnumbers, -1 to CR)  3.49%
2 CR result -> (US is tied, no outnumber bonus)  8.93%
4 CR result -> (...Attacker Outnumbers, +1 to CR) 16.51%
5 CR result -> (...Attacker Outnumbers, +1 to CR) 22.19%
6 CR result -> (...Attacker Outnumbers, +1 to CR) 21.22%
7 CR result -> (...Attacker Outnumbers, +1 to CR) 16.95%
8 CR result -> (...Attacker Outnumbers, +1 to CR)  7.09%
9 CR result -> (...Attacker Outnumbers, +1 to CR)  1.99%
=====================================================
Chance of achieving CR of 4 or more: 86.36%
=====================================================
User avatar
Banshee
Highborn
Posts: 617
Joined: Wed Oct 08, 2003 10:01 am
Location: I say go, she say yes, Dim the light, you can guess the rest!

Post by Banshee »

Just a quick question.
If I'm right your results state that a flank charge has a better CR than a rear charge, right?
2) The rear attack is nearly as good, but the Saurus have just a slightly better chance due to the extra model - even with the additional +1 to CR.

I don't think you're right here. The difference between rear and flank charges is +1 CR for chargers, +1 saurus to attack back (with 2A). So rear chargers have a CR equal to that of a flank charge with (+1 for rear charge - CR achieved by a single saurus). According to your post a saurus with 2 attacks would generat a CR of 1 or better against CoK. That's impossible! Only 50% hit, 67% wound and only 33% penetrate the knights' armour. That's less than 1!
Damnation wrote:And remember, every time you use excessive punctuation, a random brain lobe deactivates itself.
Cenyu wrote:Warhammer: Age of Reckoning: Introducing BOB ROSS as Malekith's new personal paint artist!
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

Banshee wrote:Just a quick question.
If I'm right your results state that a flank charge has a better CR than a rear charge, right?
2) The rear attack is nearly as good, but the Saurus have just a slightly better chance due to the extra model - even with the additional +1 to CR.

I don't think you're right here. The difference between rear and flank charges is +1 CR for chargers, +1 saurus to attack back (with 2A). So rear chargers have a CR equal to that of a flank charge with (+1 for rear charge - CR achieved by a single saurus). According to your post a saurus with 2 attacks would generat a CR of 1 or better against CoK. That's impossible! Only 50% hit, 67% wound and only 33% penetrate the knights' armour. That's less than 1!


The numbers are just slightly better, and it's due in part to needing 5 wounds to totally eliminate attacks back instead of 4 from the flank. This makes the CR of 4+ harder to achieve due to the slightly higher probability of being wounded back, which will in turn reduce the CR. If you look at the actual probability tables it will make more sense.

Notice that the probabilities aren't shifted by a full attack - just a few percentage points. Look at the actual probability tables. Not only does the additional defender generate an additional pair of attacks, it also makes it more probable that at least one attack from all the attackers will land, as well as making it more probable that there will be a survivor to attack back.

If you have issues with the analysis, point out the specifics please - as I said, the Saurus have a slightly better chance - I didn't state that the extra attacker was shifting the fight by a full point to CR.
User avatar
Langmann
Malekith's Tastetester & Physician
Malekith's Tastetester & Physician
Posts: 5170
Joined: Thu Oct 10, 2002 9:41 pm
Location: Putting needles into people.

Post by Langmann »

Good job I am sticking this for a while.
While running a million dollar company, singing at weddings, and his frequent jetting to Spain Elton Jon style, Dark Alliance found the time to stand on the doorstep of Games Workshop like Moses and the Pharoah and calmly state, "Let my people go."
User avatar
Thekingsfinestmen-
Highborn
Posts: 751
Joined: Mon Sep 01, 2003 7:05 pm
Location: A large ritual cementary, the Kings graveyard.

Post by Thekingsfinestmen- »

Great post man. thugh stoff.

I'd like to learn it a bit better and then use it proberbly against some opponent (bad none of my mates play lizzies :roll: ).
TKFM.
Thou Shall the Dead Come to Life Again(Ez. 37,1-14).
. . . We are waiting.
User avatar
Lord malal
Executioner
Posts: 157
Joined: Fri Jul 04, 2003 5:40 am
Location: Upstate NY, USA

Post by Lord malal »

This and all the other Lizzy posts would be great additions to the next Monthly. I think condensing all of our strates and ideas into one publication will be of great use in the coming months.

Of course for the monthly your could probably just site this link as a reference for all the tables. Just summerizing all the conclusions into one report would be suffiencent for me!
User avatar
Seekingone
Slave (off the Altar)
Posts: 24
Joined: Tue Apr 20, 2004 3:21 am

Post by Seekingone »

Greetings!

A curious thread, really :)

Zader, could you explain your calculation algorithm in more detail? I mean the one posted in the end of page 1.

I'm also a bit into probability theory, and when I make calculations for the CoK against Sauri I get different results. So one of should be wrong here :)

I don't know Perl well enough, so could you be so kind to explain your code in words?

Thanks in advance.


May Hoeth guide our ways...
SeekingOne
May Hoeth guide our ways...
SeekingOne
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

SeekingOne wrote:Greetings!

A curious thread, really :)

Zader, could you explain your calculation algorithm in more detail? I mean the one posted in the end of page 1.

I'm also a bit into probability theory, and when I make calculations for the CoK against Sauri I get different results. So one of should be wrong here :)

I don't know Perl well enough, so could you be so kind to explain your code in words?

Thanks in advance.


May Hoeth guide our ways...
SeekingOne


The short version - the code posted is for the specific instance of having one attack. I'm just using binary to represent all the to hit possibilities and adding them up based on the number of wounds. So there are 32 possible outcomes for 5 characters hitting, and in this specific case represented by counting from binary 00000 (all miss) to 11111 (all wound). The probability of one hit is additive for all the individual to-wound numbers (0-5), which are calculated in the program.

So for example to-wound of 1, p(w1) = p(00001) + p(00010) + p(00100) + p(01000) + p(10000)

Yes, it's the brute force approach, but I chose it since it's a relatively easy concept to grasp conceptually and it becomes far more difficult to construct a nice neat formula later down the line. (and definitely more complicated for the cases where the number of attacks > 1)

For each case, the probability of hitting/missing with 0 representing a chance to miss and 1 being a chance to wound. By definition the chance to miss we can assume to be 1-(chance to wound).

Chances to wound are per the combatants; the program consults the to-hit and to-wound tables. The generic chance to wound is:
(chance to hit) * (chance to wound) * (chance to fail armorsave) * (chance to fail wardsave)

For all our wounds=1 cases above, lets take an easy example saying we have a x chance to wound. The probability for w=1 cases above would then be (1-x)(1-x)(1-x)(1-x)(x).

The general case (for # attacks = n) is a bit more complicated than the n=1 case but works roughly the same way in the program. What I can't remember now however is if I had any program modifications where I should have gone back and posted updated probability tables for the CoK. Post your tables for comparison and how you are arriving at the numbers in question and we can look at the specifics.
User avatar
Eller
Trainee Warrior
Posts: 25
Joined: Sat Apr 24, 2004 5:59 pm
Location: Behind you! Mwha ha ha ha!!!
Contact:

Post by Eller »

That's pretty cool. I've always wanted to know how the Cold Blooded rule actually effects the statics, and now I have this.

Eller
Image
User avatar
Seekingone
Slave (off the Altar)
Posts: 24
Joined: Tue Apr 20, 2004 3:21 am

Post by Seekingone »

Hi again :)

Zader, thanks very much for explaining me your code.

I must say the algorithm you use is correct (at least for the simple case of 1 attack). More than that, there was a mistake on my side - in fact my results are the same, lol.

I 've also written a small app for calculating probabilities. It mainly uses the Bernulli (sp?) formula to calculate a probability of the exact m unsaved wounds caused by n attacks. Just in case you didn't know it, the formula looks like this:

n!/m!(n-m)! * pow(p, m) * pow((1-p), n-m)

To clarify the notation, in the formula above:

"x!" means "x factorial" (multiplication of all natural numbers from 1 to x; x! = 1*2*3...*x)

pow(x, y) means "x raised to the y-th power" (not sure if this is correct english phrase, but hope you get what I mean :))

n - number of attacks
m - exact number of unsaved wounds
p - probability of a single unsaved wound

As you see, this formula enables you to make one universal program that would make calculations for any combination of the three initial parameters that you enter.

My program also has an algorithm that calculates probabilities for wounds caused by the defenders striking back. The algorithm is based on the "full probability" formula, which works generally similar to the way you used in your calculations above. Unfortunately, my program only handles the simplest case of 1A 1W models in the defending unit.

I have to say, your calculations inspired me to make some enhancements in my app... If I succeed with them, I'll post my results for comparison :)

By the way, your calculations of Lisardmen Ld tests are absolutely correct :) You even helped me to find my own mistake in the similar calculations of mine ;)


May Hoeth guide our ways...
SeekingOne
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

SeekingOne,

The alogrithm works for the n>1 cases as well, it just uses a base n counting method which is handled by the program. (So binary for 2 attacks, base 3 for 3 attacks, etc.)

The two methods we are using are equivilent for the probability models of the individual units; the main reason I'm using the long way is that it's easier to grasp conceptually. I remember doing the proof to the formula you listed many, many years ago in college ... but I didn't want to have to try and explain it here. ;) That and it took me 15 minutes to write the necessary code, and it would be an all day endeavor to try and dig out some of my college textbooks. :)
User avatar
Alkha'gthor
Corsair
Posts: 87
Joined: Mon Mar 17, 2003 1:56 am

Post by Alkha'gthor »

It's astounding the formulas that we have derived through the years for our maths most of the stuff that I've seen so far and worked out I have done on a basic calculator and the rest in my head, amazing how the maths works it self out in your head. That formula doesn't look quite the same as the one we used in probability it may have a different name and notation, where did you do your studies SeekingOne and Zader if you don't mind me asking?
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
User avatar
Seekingone
Slave (off the Altar)
Posts: 24
Joined: Tue Apr 20, 2004 3:21 am

Post by Seekingone »

@Zader
I see :) Sure, it also took me quite some time to revive my old university knowledge.
And don't get me wrong, I didn't mean to doubt your Prob Theory competentency. On contrary, it looks pretty impressive to me.

Now, to business :)
I took my time and modified the app I used so that now it is capable to calculate the "total" effective probabilities of causing not less than M wounds for an attacking unit with several different types of attacks (like knights and their mounts). Basically it uses the same principle as you used for calculating the P of CoK causing not less than 3 wounds.

Here are the results I got.
The calculation is done for the same case as above: CoK charging Sauri, 5 attacks of knights, 5 attacks of mounts.

Below is the probability table which takes into account both knights and riders (P is in %)

Not less than 0 Wounds : 100.00 (obviously :D)
Not less than 1 Wounds : 96.11
Not less than 2 Wounds : 80.66
Not less than 3 Wounds : 53.76
Not less than 4 Wounds : 26.84
Not less than 5 Wounds : 9.71
Not less than 6 Wounds : 2.48
Not less than 7 Wounds : 0.43
Not less than 8 Wounds : 0.05
Not less than 9 Wounds : 0.00
Not less than 10 Wounds : 0.00

This table shows a result for "Not less thjan 3 wounds" which differs from the one you got above (66.91%). Let's check it:

P(>=3 wounds) = 1 - P(<3 wounds)

P(<3 wounds) =
p(Knights get 0 wounds * CO get 0 wounds) +
p(Knights get 1 wounds * CO get 0 wounds) +
p(Knights get 0 wounds * CO get 1 wounds) +
p(Knights get 2 wounds * CO get 0 wounds) +
p(Knights get 0 wounds * CO get 2 wounds) +
p(Knights get 1 wounds * CO get 1 wounds) =
0,1 * 0,4 +
0,1 * 0,4 +
0,29 * 0,4 +
0,1 * 0,16 +
0,34 * 0,4 +
0,29 * 0,4 = 0,464

P(>=3 wounds) = 1 - P(<3 wounds) = 1 - 0,464 = 0,536
This is the same as in the table, with rounding difference. So, unless there's something wrong with my logic here, it looks like it's even harder to break Sauri than it seemed.

Finally, just for fun of it, let's have a look at what benefit do we get from upgrading 1 knight to a champ. Basically this means +1A for knights.

New table of exact probabilities for knights looks as follows (P is in %):

0 Wounds : 6.10
1 Wounds : 21.73
2 Wounds : 32.27
3 Wounds : 25.55
4 Wounds : 11.38
5 Wounds : 2.70
6 Wounds : 0.27

For CO it remains the same.

New table of "effective" probabilities (P is in %):

Not less than 0 Wounds : 100.00
Not less than 1 Wounds : 97.56
Not less than 2 Wounds : 86.42
Not less than 3 Wounds : 63.78
Not less than 4 Wounds : 36.87
Not less than 5 Wounds : 16.09
Not less than 6 Wounds : 5.17
Not less than 7 Wounds : 1.20
Not less than 8 Wounds : 0.19
Not less than 9 Wounds : 0.02
Not less than 10 Wounds : 0.00
Not less than 11 Wounds : 0.00

The difference looks quite noticeable to me, especially for the two most likely results: >=2 (80,66 vs 86,42) and >=3 (53,76 vs 63,78).
Conclusion: CoK champion really looks like a worthy upgrade ;)

Well, hope someone finds this boring stuff useful :)


May Hoeth guide our ways...
SeekingOne
Last edited by Seekingone on Tue Apr 27, 2004 7:07 am, edited 1 time in total.
User avatar
Thee forsaken one
Forsaken Master of Fallen Gods
Posts: 313
Joined: Fri Jan 03, 2003 8:39 pm
Location: Near Glasgow, SCOTLAND
Contact:

Post by Thee forsaken one »

Although very interesting and I'm sure accurate all this is assuming that:

1. You can get your unit into a good position.
2. For frontal charge I'm asuming you are not counting the unit as having spears
3. The lizzie general has no back up plan for your units.
4. Have you tried putting a Saurus hero in the unit and a noble in the other to see how they affect it?

However the results do look around about accurate for cold blooded checks and it is true that making lizzies break is hard.
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

Thee Forsaken One wrote:Although very interesting and I'm sure accurate all this is assuming that:

1. You can get your unit into a good position.
2. For frontal charge I'm asuming you are not counting the unit as having spears
3. The lizzie general has no back up plan for your units.
4. Have you tried putting a Saurus hero in the unit and a noble in the other to see how they affect it?

However the results do look around about accurate for cold blooded checks and it is true that making lizzies break is hard.


1) Except for units with Huanchi's Blessed Totem, or unless you charge with your JSoD, one advantage the Druchii do have is movement. It's very seldom that the Saurus are going to get the charge. Notice that I don't say that they won't win even if they are charged...
2) I'm not using spears for these examples no - primarily since I'm mostly modeling flank charges there's no benefit to spears. For almost all Druchii units, a frontal charge against a Saurus block is going to be either a last resort, or accompanied by a simultaneous flank charge assuming MSE/MSU.
3) This is sort of an infinitely looping argument ... who's to say we don't have backup plans for your backup plans? :P
4) No, I'm avoiding character's for a reason - there's just too many combinations of magic items, weapons, armor, and characters to be worth trying to model. Magic item combinations are frequently discussed on a number of posts on the various boards - the probability models are just unit vs. unit.
Zader
Warrior
Posts: 64
Joined: Mon Mar 29, 2004 2:22 pm
Location: Right behind you ...

Post by Zader »

SeekingOne wrote:@Zader

Below is the probability table which takes into account both knights and riders (P is in %)

Not less than 0 Wounds : 100.00 (obviously :D)
Not less than 1 Wounds : 96.11
{snip}


We get the same results for the knights+CO- my table is for the exact probability of x number of wounds, while your table shows the probability of that many wounds or less. To equate the results just add the lower number of wounds together - the numbers work out to be the same that way.

This table shows a result for "Not less thjan 3 wounds" which differs from the one you got above (66.91%). Let's check it:

Nope, It's around 53.3% - check the table again for P(3 -> 10) and add them together. This matches the numbers you have below (with room for some rounding differences).
{snip}
.
.
P(>=3 wounds) = 1 - P(<3 wounds) = 1 - 0,464 = 0,536
This is the same as in the table, with rounding difference. So, unless there's something wrong with my logic here, it looks like it's even harder to break Sauri than it seemed.

Finally, just for fun of it, let's have a look at what benefit do we get from upgrading 1 knight to a champ. Basically this means +1A for knights.
{snip}
The difference looks quite noticeable to me, especially for the two most likely results: >=2 (80,66 vs 86,42) and >=3 (53,76 vs 63,78).
Conclusion: CoK champion really looks like a worthy upgrade ;)

Well, your math looks right to me at a glance. :) The Champion shifts the results by several percentage points, but not by a whole +1 to CR, sort of as we expect. I always take one myself when I run Cold One Knights.

Well, hope someone finds this boring stuff useful :)


Hey, it certainly beats working right? :)
Mightypeon
Trainee Warrior
Posts: 35
Joined: Tue Feb 03, 2004 6:22 pm
Location: Berlin

Post by Mightypeon »

Some Additions:
1: All examples assumed onmarked Saurus Warriors.
In my experience, the marking of Quetzl is the most propable.
The stats are signifcantly tweaked as the LZM Chance to pass armour saves agasint CoKs doubles.
In addition, some posters had very strange situations which do not happen in a normal battle field (unless you are playing CoS).

A) The CoKs in the Flank situation: This is rare. The only way this is goign to happen is by breaking the LzM flanks, these flanks tend to be protected by Skink+ Kroxigor formations.
A more propably situation is 6 CoKs + Champ in the Front, 5 Dark Riders + Musican in the Flank ov a Max 20 Saurus unit.
I will just present some averages:
6 CoK attacks will hit 4 times, than they have to wound on a 3, the Blessed Saurus warriors get their save on a 5+. It equates to roughly 1.7 dead Saurus warriors.
The mounts have 5 attacks, hitting wounding and penetrating on 50%50 chances. They will kill roughly 0.6 Saurus warriors.
We can assume that the CoK charge will kill 2 Saurus on average, 3 if you are a bit lucky.
Now the Dark Riders:
I assume that they have spears.
They have 5 attacks hitting on 3s wounding on 4 penetrating on 4s. Together with their mounts they will kill another Saurus.
This means: 2 Saurus Warriors can strike back agasint the Dark Riders
1-2 Saurus Warriors can strike back agasint the CoK
1 Saurus Warriors may choose wether he hits the Dark Riders or the CoK.

First assumption:
Druchii got lucky, 3 Saurus Warriors died in the front charge, on in the Side.
The Saurus champion stirkes back agasint the CoK. He will kill 0.333 CoKs.
We assume that he kills nothing in this scenario.
The Side rank stikes back: 3 Saurus warrios can strike, I assume that every remotly decent LzM general will attack the Dark Riders with his edge warrior.
The 6 attacks will kill 1.6666 Dark Riders on an average.
As we rounded down in the last example we now round up, this leads to 2 kills.
CR:
De unit size=5 CoK + 3 DR, 16
Saurus unit Size= 16
Draw.
DE CR: 3s kill in the front, on kill at the flank, a flank, a standart. 6
Sarus CR: Standart, 2 kills=3
The Lizards loose by 3 meaning that they roll on a 5. Roughly 50%50 odds for the DE. This is in case they got lucky.

If the whole things runs normal the DE end up with one kill less, meaning that they have only a CR of 5 while the LZM CR is boosted to 4 as they now get outnumbering.
It is very likely that the LZM pass the LD on a 7.
If the LZM get lucky they end up with an additional kill agasint the CoKs, further upping their CR to 5. The Druchii will stay at 5, meaing that a Draw is the outcome.

This analys should prvoe that the blessed Spawning of Quetzl is something to watch out for.



Inregards to a Hydra banner, if you have a Battle standart I have a Scar vet who has 4 S7 attacks. In addition, You Battle standart bearer is a significantly softer target that the reast of the unit, so that I will get an additional dealt wound agasint him.
Taking your own characters into account while not considering the significantly stronger LZM ones is self illusion.
User avatar
The chosen
Assassin
Posts: 503
Joined: Thu Jan 08, 2004 12:53 pm
Location: In the cold Denmark

Post by The chosen »

take the lore of shadows for your mages... then cast shades of death so that you cause fear and make sure to outnumber them...
ALL YOUR BASE ARE BELONG TO US
Locked