I’ve lately started playing Warhammer after a pretty prolonged absence. To get back into it and to get use to the new rules I played a couple of small skirmishes, orcs against high elves, and I noticed something odd. Very often when my block of orc spearmen would come up against a smaller block of high elf spearmen it was the elves who would break.

This seemed a bit odd to me since it seemed as though the odds should all be in favor of the elves. With smaller bases they could have 6 elves come into contact with 5 orcs. Secondly their spearmen have a big advantage in that they are allowed an extra row of attacks, meaning that they would get 18 attacks to my 10. Lastly the high elves had a better leadership score.

At first I wrote off the success of the orcs to a combination of lucky rolls and to getting my opponent to charge, reducing the number of attacks the elves could make. That only got me so far, though. I couldn’t recall making any really spectacular rolls and as I thought about it I realized that more than a few of those breaks occurred when I charged, reducing my number of attacks, or in the second round of combat when three rows of spearmen were able to fight.

I thought maybe the orcs just had a significantly higher probability of wounding the elves. Their high toughness meant the elves needed to roll a 5 to wound and even though they needed only a 3 to hit I knew that their odds were worse than the 4 to hit and 4 to wound the orcs needed. However after I thought about it a little longer I realized the odds weren’t really that different. For the orcs the odds of getting a hit and wounding were 1 in 4. For the elves the odds were 2 in 9. A little in favor of the orcs until you give the elves nearly twice as many attacks. Armor saves for the two units was identical so that was no help.

Time to start digging down into the exact probabilities and to stop guessing.

The first thing I had to do was to calculate the odds of each race causing a casualty. To cause a casualty there are three successive rolls involved. First the roll to hit. Next the roll to wound. Finally the armor save. If any one of those rolls fails (or succeeds in the case of armor) then the attack fails to cause a casualty. This makes the odds of causing a casualty fairly easy to calculate. For the orcs the odds are 1 in 2 times 1 in 2 times 2 in 3, or 1 in 6. For the elves the odds of causing a wound are 2 in 3 time 1 in 3 time 2 in 3 or 4 in 27.

Now that I had the odds of a single attack causing a casualty I could work out the odds of all the casualties each unit could cause. For a specific number of casualties the odds would be the odds of success raised to the number of casualties time the odds of failure raised to the remaining units times the number of possible combinations that resulted in that exact number of casualties. Put into a formula this is S^{c}*(1-S)^{(t-c)}*C(t,c) where S was the chance of success, c was the number of casualties caused, t was the total number of attacks, and C() was the combination formula of t!/c!(t-c)!.

For the orcs this gave me the following table:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

.1615 | .3230 | .2907 | .1550 | .0543 | .0130 | .0022 | .0002 | .0000 | .0000 | .0000 |

For the elves this gave me this table:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

.0558 | .1746 | .2582 | .2395 | .1562 | .0760 | .0287 | .0085 | .0020 | .0004 | .0001 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

Crossing the two tables together gave me the chance of each combination occurring. As an example since there is a 32.30% chance of the orcs inflicting one casualty and a 25.82% chance of the elves inflicting 2 there is an 8.34% (.3230*.2582) chance that the orcs will inflict 1 casualty while the elves will inflict 2.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0 | .0090 | .0180 | .0162 | .0086 | .0030 | .0007 | .0001 | .0000 | .0000 | .0000 | .0000 |

1 | .0282 | .0564 | .0508 | .0271 | .0095 | .0023 | .0004 | .0000 | .0000 | .0000 | .0000 |

2 | .0417 | .0834 | .0751 | .0400 | .0140 | .0034 | .0006 | .0001 | .0000 | .0000 | .0000 |

3 | .0387 | .0773 | .0696 | .0371 | .0130 | .0031 | .0005 | .0001 | .0000 | .0000 | .0000 |

4 | .0252 | .0504 | .0454 | .0242 | .0085 | .0020 | .0003 | .0000 | .0000 | .0000 | .0000 |

5 | .0123 | .0246 | .0221 | .0118 | .0041 | .0010 | .0002 | .0000 | .0000 | .0000 | .0000 |

6 | .0046 | .0093 | .0083 | .0044 | .0016 | .0004 | .0001 | .0000 | .0000 | .0000 | .0000 |

7 | .0014 | .0028 | .0025 | .0013 | .0005 | .0001 | .0000 | .0000 | .0000 | .0000 | .0000 |

8 | .0003 | .0007 | .0006 | .0003 | .0001 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

9 | .0001 | .0001 | .0001 | .0001 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

10 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

11 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

12 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

13 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

14 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

15 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

16 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

17 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

18 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

I also constructed two more tables. Each table lists the chance of either the orcs or elves failing a leadership roll under a specific set of circumstances.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0 | |||||||||||

1 | |||||||||||

2 | |||||||||||

3 | .4167 | ||||||||||

4 | .5833 | .4167 | |||||||||

5 | .7222 | .5833 | .4167 | ||||||||

6 | .8333 | .7222 | .5833 | .4167 | |||||||

7 | .9167 | .8333 | .7222 | .5833 | .4167 | ||||||

8 | .9722 | .9167 | .8333 | .7222 | .5833 | .4167 | |||||

9 | .9722 | .9722 | .9167 | .8333 | .7222 | .5833 | .4167 | ||||

10 | .9722 | .9722 | .9722 | .9167 | .8333 | .7222 | .5833 | .4167 | |||

11 | .9722 | .9722 | .9722 | .9722 | .9167 | .8333 | .7222 | .5833 | .4167 | ||

12 | .9722 | .9722 | .9722 | .9722 | .9722 | .9167 | .8333 | .7222 | .5833 | .4167 | |

13 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9167 | .8333 | .7222 | .5833 | .4167 |

14 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9167 | .8333 | .7222 | .5833 |

15 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9167 | .8333 | .7222 |

16 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9167 | .8333 |

17 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9167 |

18 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0 | .4167 | .5833 | .7222 | .8333 | .9167 | .9722 | .9722 | .9722 | .9722 | .9722 | .9722 |

1 | .2778 | .4167 | .5833 | .7222 | .8333 | .9167 | .9722 | .9722 | .9722 | .9722 | .9722 |

2 | .2778 | .4167 | .5833 | .7222 | .8333 | .9167 | .9722 | .9722 | .9722 | .9722 | |

3 | .2778 | .4167 | .5833 | .7222 | .8333 | .9167 | .9722 | .9722 | .9722 | ||

4 | .2778 | .4167 | .5833 | .7222 | .8333 | .9167 | .9722 | .9722 | |||

5 | .2778 | .4167 | .5833 | .7222 | .8333 | .9167 | .9722 | ||||

6 | .2778 | .4167 | .5833 | .7222 | .8333 | .9167 | |||||

7 | .2778 | .4167 | .5833 | .7222 | .8333 | ||||||

8 | .2778 | .4167 | .5833 | .7222 | |||||||

9 | .2778 | .4167 | .5833 | ||||||||

10 | .2778 | .4167 | |||||||||

11 | .2778 | ||||||||||

12 | |||||||||||

13 | |||||||||||

14 | |||||||||||

15 | |||||||||||

16 | |||||||||||

17 | |||||||||||

18 |

The first table is for the orcs and the second is for the elves. These tables were built with the assumption that the general (a hero in this case) is close enough to allow the unit to use his leadership skill. It also assumes a 2 point advantage in the combat results for the orcs for extra ranks (4 ranks instead of 3) and superior numbers (these are conditions under which the combats were occurring). Because orcs cost so much less than high elves they will probably almost always have superior numbers, though the elves can certainly field a fourth rank.

An example of the tables in action is if the orcs inflict 1 casualties and the elves inflict 2 the elves need an 8 (9 for their leadership-1 for the combat results) for their leadership roll, which is a 27.78% chance to break.

Multiplying these two tables with the probability table for a given combination gave me my last two tables. Again the first table is for orcs and the second table is for elves. They show the odds that in a combat a given combination of wounds will occur and that the loosing side (if there is one) will fail to make its leadership roll. As a final example since there is an 8.34% chance of the orcs inflicting 1 casualty and the elves inflicting 2 in a combat and this will result in a break for the elves 27.78% of the time there is a 2.32% (.0834*.2778) chance that in a combat the orcs will inflict 1 casualty, the elves will inflict 2, and the elves will break.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0 | |||||||||||

1 | |||||||||||

2 | |||||||||||

3 | .0161 | ||||||||||

4 | .0147 | .0210 | |||||||||

5 | .0089 | .0143 | .0092 | ||||||||

6 | .0039 | .0067 | .0049 | .0019 | |||||||

7 | .0013 | .0023 | .0018 | .0008 | .0002 | ||||||

8 | .0003 | .0006 | .0005 | .0002 | .0001 | .0000 | |||||

9 | .0001 | .0001 | .0001 | .0001 | .0000 | .0000 | .0000 | ||||

10 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | |||

11 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | ||

12 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | |

13 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

14 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

15 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

16 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

17 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

18 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 | .0000 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0 | .0038 | .0105 | .0117 | .0072 | .0028 | .0007 | .0001 | .0000 | .0000 | .0000 | .0000 |

1 | .0078 | .0235 | .0296 | .0196 | .0079 | .0021 | .0004 | .0000 | .0000 | .0000 | .0000 |

2 | .0232 | .0313 | .0233 | .0101 | .0028 | .0005 | .0001 | .0000 | .0000 | .0000 | |

3 | .0193 | .0155 | .0076 | .0023 | .0004 | .0001 | .0000 | .0000 | .0000 | ||

4 | .0067 | .0035 | .0012 | .0002 | .0000 | .0000 | .0000 | .0000 | |||

5 | .0011 | .0004 | .0001 | .0000 | .0000 | .0000 | .0000 | ||||

6 | .0001 | .0000 | .0000 | .0000 | .0000 | .0000 | |||||

7 | .0000 | .0000 | .0000 | .0000 | .0000 | ||||||

8 | .0000 | .0000 | .0000 | .0000 | |||||||

9 | .0000 | .0000 | .0000 | ||||||||

10 | .0000 | .0000 | |||||||||

11 | .0000 | ||||||||||

12 | |||||||||||

13 | |||||||||||

14 | |||||||||||

15 | |||||||||||

16 | |||||||||||

17 | |||||||||||

18 |

Totaling the contents of each table then tells me exactly what the odds are that in a given combat the side will break. In the case of the orcs it winds up being an 11.00% chance they will break while in the case of the high elves if winds up being a much higher 27.77% chance they will break.

Since I was pulling most of this math out of my butt (surprisingly there’s a lack of books on calculating break odds for WFB) I was a bit suspicious of these numbers. I never took a class in statistics or probability though I knew I had a pretty good grasp of the concepts and I haven’t taken a math class in over 20 years. So I sat down and wrote a little program for my calculator that would run a number of battles and give me the results. After simulating 30,000 combats I came up with 3,324 breaks for the orcs, 8,287 breaks for the elves, and 18,389 combats where neither side broke. This was against a projected result of 3,300 breaks for the orcs, 8,330 breaks for the elves, and 18,370 combats with no breaks occurring. That looked pretty good to me and after a quick spot of research and feeding it through a chi^{2} goodness of fit test I came up with a 81.21% confidence. Research said that was to be considered extremely high confidence but since I don’t do chi^{2} tests that often I wanted a little more data. I left the results as they were but changed the predicted values to 3,300/8,180/18,520. This was the predicted result if the predicted odds for the high elves, who already had scored fewer breaks than predicted, was half a percent lower (27.27% instead of 27.77%). Confidence dropped to 28.64%. Seeing the predicted results coming out so close to the observed results made me very confident that the theories I was using were correct.

Of course back when I made the combinations table I already knew I didn’t want to sit down with a calculator and calculate the contents of all 180 combinations so I set up a spreadsheet to do it, and the following steps, for me. The totals surprised me so I decided to change a couple of values and see what that did to the results. After about the third time of editing some of the tables I decided it would be better to just put the values where I could easily change them and to upgrade the code to make building the tables more automatic. With that done I could quickly and easily change values and see the odds of breaking for each side if the orcs charged, if the elves charged, if the elves had another rank, etc. and I found my answer; the single biggest factor in the odds of who broke was the combat result bonus. Add a fourth row of elves and reduce the orc bonus from +2 to +1 and now there was a 21.52% chance of the orcs breaking and only a 16.12% chance of the elves breaking. Nothing else had as profound an effect. If the orc spearmen charged then the odds would swing to be very slightly in the elves favor (17.83% to 16.79% orcs to elves), if the orcs gave up spears and shields for two choppas the odds shifted only a little in the first round and more in the second but the elves were still at a disadvantage (and now the orcs could charge so it was a trade off of risking breaking on the first round for charging spearmen or breaking in the second round for choppas) but really nothing had the impact of that extra rank.

So I guess the moral is “Rank up. It has a bigger effect than you probably realize.”And just in case you want to play around with the numbers yourself here is a link to the spreadsheet. So that it is accessible to as many people as possible without having to use pirated software it is in an OpenDocument Spreadsheet (.ods) format which can be read with OpenOffice and a variety of other spreadsheet programs.