Combat Table Probabilities

[Matthew]

Just wanted to outline the probabilities of scoring hits here:


Light Foot versus:

Type	Dice	Probability
Light Foot	1 Die per 1 Figure, 6+	12-in-72
Heavy Foot	1 Die per 2 Figures, 6+	6-in-72
Armoured Foot	1 Die per 3 Figures, 6+	4-in-72
Light Horse	1 Die per 2 Figures, 6+	6-in-72
Medium Horse	1 Die per 3 Figures, 6+	4-in-72
Heavy Horse	1 Die per 4 Figures, 6+	3-in-72

Heavy Foot versus:

Type	Dice	Probability
Light Foot	1 Die per 1 Figure, 5-6+	24-in-72
Heavy Foot	1 Die per 1 Figure, 6+	12-in-72
Armoured Foot	1 Die per 2 Figures, 6+	6-in-72
Light Horse	1 Die per 2 Figures, 6+	6-in-72
Medium Horse	1 Die per 3 Figures, 6+	4-in-72
Heavy Horse	1 Die per 4 Figures, 6+	3-in-72

Armoured Foot versus:

Type	Dice	Probability
Light Foot	1 Die per 1 Figure, 4-6+	36-in-72
Heavy Foot	1 Die per 1 Figure, 5-6+	24-in-72
Armoured Foot	1 Die per 1 Figure, 6+	12-in-72
Light Horse	1 Die per 1 Figure, 6+	12-in-72
Medium Horse	1 Die per 2 Figures, 6+	6-in-72
Heavy Horse	1 Die per 3 Figures, 6+	4-in-72

Light Horse versus:

Type	Dice	Probability
Light Foot	2 Dice per 1 Figure, 5-6+	48-in-72
Heavy Foot	2 Dice per 1 Figure, 6+	24-in-72
Armoured Foot	1 Die per 1 Figure, 6+	12-in-72
Light Horse	1 Die per 1 Figure, 6+	12-in-72
Medium Horse	1 Die per 2 Figures, 6+	6-in-72
Heavy Horse	1 Die per 3 Figures, 6+	4-in-72

Medium Horse versus:

Type	Dice	Probability
Light Foot	2 Dice per 1 Figure, 4-6+	72-in-72
Heavy Foot	2 Dice per 1 Figure, 5-6+	48-in-72
Armoured Foot	2 Dice per 1 Figure, 6+	24-in-72
Light Horse	1 Die per 1 Figure, 5-6+	24-in-72
Medium Horse	1 Die per 1 Figure, 6+	12-in-72
Heavy Horse	1 Die per 2 Figures, 6+	6-in-72

Heavy Horse versus:

Type	Dice	Probability
Light Foot	4 Dice per 1 Figure, 5-6+	96-in-72
Heavy Foot	3 Dice per 1 Figure, 5-6+	72-in-72
Armoured Foot	2 Dice per 1 Figure, 5-6+	48-in-72
Light Horse	2 Dice per 1 Figure, 5-6+	48-in-72
Medium Horse	1 Die per 1 Figure, 5-6+	24-in-72
Heavy Horse	1 Die per 1 Figure, 6+	12-in-72

I think that I worked out the probabilities correctly, but am open to corrections! Anyway, it is interesting to see which methods work out to have the same average:

1 Die per 1 Figure, 5-6+ = 2 Dice per 1 Figure, 6+ = 24-in-72
2 Dice per 1 Figure, 4-6+ = 3 Dice per 1 Figure, 5-6+ = 72-in-72

Interesting to see that Armoured Foot versus Light Foot, and Light Foot versus Horse seem out of whack with the general probability progressions. Could be that Light Foot is viewed as better able to take on horse than other forms of foot, and that Armoured Foot is less able to deal with Light Foot than other speedier units.

[cooper]

Cool. Looking forward to your further thoughts.

[Matthew]

Ta, it is worth noting that under this paradigm the combat capability advancement of the classes in OD&D make a lot of sense. That is to say:

1d6+1 = 2d6, on average, when the probability of scoring a hit is 6+.

Assuming that is the case, we can restate the combat progression of the fighter as:

Level	Fighting Capability	Probability
1	1+1	24-in-72
2	2+1	36-in-72
3	3 or 4−1	36-in-72
4	4	48-in-72
5	4+1 or 5	60-in-72
6	4+1 or 6	60 or 72-in-72
7	8−1	84-in-72
8	8	96-in-72
9	8+1	108-in-72
10	8+1	108-in-72

When the probability of scoring a hit is 5+ it looks more like:

Level	Fighting Capability	Probability
1	1+1	36-in-72
2	2+1	60-in-72
3	3 or 4−1	72 or 84-in-72
4	4	96-in-72
5	4+1 or 5	108 or 120-in-72
6	4+1 or 6	108 or 144-in-72
7	8−1	180-in-72
8	8	192-in-72
9	8+1	204-in-72
10	8+1	204-in-72

And when the probability of scoring a hit is 4+ it is:

Level	Fighting Capability	Probability
1	1+1	48-in-72
2	2+1	84-in-72
3	3 or 4−1	108 or 120-in-72
4	4	144-in-72
5	4+1 or 5	156 or 180-in-72
6	4+1 or 6	156 or 216-in-72
7	8−1	276-in-72
8	8	288-in-72
9	8+1	300-in-72
10	8+1	300-in-72

[Deleted]

The actual chance of either of two dice scoring a "6" is 22 - in - 72 not 24. The chance that both dice score a "6" actually works against the overall odds that either will score a six.

Getting a "6" on either 2D6 = (1/6 + 1/6) - (1/6 x 1/6)

I may be a little rusty but I think this is correct.

Also, what if the +1 in fighting capability was an extra die to roll (as a magic sword in mass combat) and not an addition to the die score?

[Matthew]

Jun 9, 2011 15:15:26 GMT -6 @rabbit said:

The actual chance of either of two dice scoring a "6" is 22 - in - 72 not 24. The chance that both dice score a "6" actually works against the overall odds that either will score a six.

Getting a "6" on either 2D6 = (1/6 + 1/6) - (1/6 x 1/6)

I may be a little rusty but I think this is correct.

Right, the above just gives the average number of hits over 72 attacks, as if we would only count the double six result as one hit if we were only interested in the binary result of 0 hits and 1+ hits.

Jun 9, 2011 15:15:26 GMT -6 @rabbit said:

Also, what if the +1 in fighting capability was an extra die to roll (as a magic sword in mass combat) and not an addition to the die score?

Depends on what the core is needed to hit. If you need 6+ then +1 to the die roll is the same as +1 die in terms of average hits. If you need 5+ then +1 die is twice as good as +1 to hit on one die. It is unlikely that 4 men +1 refers to five dice, though.

[Matthew]

Been looking more closely at the averages in Chain Mail and how they intersect with OD&D recently. Here are two discussion threads that relate back to this one:

[CM] A Greyhawk Scenario
Chain Mail to Dungeons & Dragons

[kent]

Matthew, Id like to follow what you are doing here and on K&KA but want to read Chainmail properly first. Im fairly sure you keep organised bookmarks on certain thread topics, can you recommend a thread or two which would serve as a good companion for a thorough reading with as little contradiction and foolishness as possible. I know there are basic ambiguities in the text and would like to see them resolved.

[Matthew]

You know, that is a good question and I am not sure I did bookmark any of the threads discussing Chain Mail. Probably because they are mostly confined to this subforum or the OD&D Knights & Knaves OD&D subforum, which moves rather slowly. The best discussions I have had about the practical implementation of the game were probably with "ChigagoWiz" when he was running some scenarios. Now that I think of it, his blog had some interesting discussion and I have some summary documents ChigagoWiz put together. I will have a hunt around this weekend and see if I can find some useful links for you.

[kent]

Thanks. Yes I'll find the chicagoWiz thread - I saw that before. I think it had images and examined even simple concepts carefully. Its just that Im not interested in the debates so much as insights and conclusions which simplify a reading of the text. Ta.

[Matthew]

In case you did not manage to locate them, here are a couple of the threads in question:

Chainmail Play by Play and BtB questions...
Chainmail - resolving missile fire

[kent]

Thanks matthew.

[Matthew]

No problem; if I think of any other threads, or stumble upon them, I will link them back to you.

I was looking at the tables again today and it occurred to me that there are two visible trends worth noting:

Foot versus Foot (Light/Medium/Heavy)

Light Foot: 6/36, 3/36, 2/36
Medium Foot: 12/36, 6/36, 3/36
Heavy Foot: 18/36, 12/36, 6/36

Horse versus Horse (Light/Medium/Heavy)

Light Horse: 6/36, 3/36, 2/36
Medium Horse: 12/36, 6/36, 3/36
Heavy Horse: 24/36, 12/36, 6/36

The odd one out being Heavy Horse versus Light Horse, 24/36 instead of 18/36.

Horse versus Foot (Light/Medium/Heavy)

Light Horse: 24/36, 12/36, 6/36
Medium Horse: 36/36, 24/36, 12/36
Heavy Horse: 48/36, 36/36, 24/36

In other words shifted one theoretical class up for each equivalent foot unit (Light Horse fights as Medium Foot) and then doubled.

Foot versus Horse is where it gets tricky, because light foot and medium foot have equal defensive effectiveness, and heavy foot defend one class better. It would make a lot of sense if originally the classes fought at the following levels of equivalence:

Light Foot
Medium Foot = Light Horse (double dice against foot)
Heavy Foot = Medium Horse (double dice against foot)
Heavy Horse (double dice against foot)

That would yield the following tables:

Light Foot versus:

Type	Dice	Probability
Light Foot	1 Die per 1 Figure, 6+	12-in-72
Medium Foot	1 Die per 2 Figures, 6+	6-in-72
Heavy Foot	1 Die per 3 Figures, 6+	4-in-72
Light Horse	1 Die per 2 Figures, 6+	6-in-72
Medium Horse	1 Die per 3 Figures, 6+	4-in-72
Heavy Horse	1 Die per 4 Figures, 6+	3-in-72

Medium Foot versus:

Type	Dice	Probability
Light Foot	1 Die per 1 Figure, 5+	24-in-72
Medium Foot	1 Die per 1 Figure, 6+	12-in-72
Heavy Foot	1 Die per 2 Figures, 6+	6-in-72
Light Horse	1 Die per 1 Figure, 6+	12-in-72
Medium Horse	1 Die per 2 Figures, 6+	6-in-72
Heavy Horse	1 Die per 3 Figures, 6+	4-in-72

Heavy Foot versus:

Type	Dice	Probability
Light Foot	1 Die per 1 Figure, 4+	36-in-72
Medium Foot	1 Die per 1 Figure, 5+	24-in-72
Heavy Foot	1 Die per 1 Figure, 6+	12-in-72
Light Horse	1 Die per 1 Figure, 5+	24-in-72
Medium Horse	1 Die per 1 Figure, 6+	12-in-72
Heavy Horse	1 Die per 2 Figures, 6+	6-in-72

Light Horse versus:

Type	Dice	Probability
Light Foot	2 Dice per 1 Figure, 5+	48-in-72
Medium Foot	2 Dice per 1 Figure, 6+	24-in-72
Heavy Foot	1 Die per 1 Figure, 6+	12-in-72
Light Horse	1 Die per 1 Figure, 6+	12-in-72
Medium Horse	1 Die per 2 Figures, 6+	6-in-72
Heavy Horse	1 Die per 3 Figures, 6+	4-in-72

Medium Horse versus:

Type	Dice	Probability
Light Foot	2 Dice per 1 Figure, 4+	72-in-72
Medium Foot	2 Dice per 1 Figure, 5+	48-in-72
Heavy Foot	2 Dice per 1 Figure, 6+	24-in-72
Light Horse	1 Die per 1 Figure, 5+	24-in-72
Medium Horse	1 Die per 1 Figure, 6+	12-in-72
Heavy Horse	1 Die per 2 Figures, 6+	6-in-72

Heavy Horse versus:

Type	Dice	Probability
Light Foot	2 Dice per 1 Figure, 3+	96-in-72
Medium Foot	2 Dice per 1 Figure, 4+	72-in-72
Heavy Foot	2 Dice per 1 Figure, 5+	48-in-72
Light Horse	1 Die per 1 Figure, 4+	36-in-72
Medium Horse	1 Die per 1 Figure, 5+	24-in-72
Heavy Horse	1 Die per 1 Figure, 6+	12-in-72

Red highlights where I have changed the dice, and whether the average probabilities have changed as a result.

Not only does this address the weakness of footmen in the game and make the original point suggestions more reasonable, but it is also more aesthetically pleasing in its symmetry.

In addition, it suggest an interesting paradigm for how armour classes might have been derived for OD&D:

Foot	Horse	Armour Class	Armour Type	Shield
Light/Light		8	None	Yes
Light/Medium		7	Leather	No
Medium/Medium	Light/Light	6	Leather	Yes
Medium/Heavy	Light/Medium	5	Mail	No
Heavy/Heavy	Medium/Medium	4	Mail	Yes
	Medium/Heavy	3	Plate	No
	Heavy/Heavy	2	Plate	Yes

Given that fatigue and flank attacks allow for troops at the high and low end of the spectrum to attack at higher (Heavy Horse +1/Heavy Horse) and lower classes (Light Foot −1/Light Foot −1), this could account for the full 1-9 range, and even potentially the elusive reason for the introduction of AC 10. Obviously, the rationale for combining both attack and defence values into armour class would be explained by the fact that normal men all have THAC0 19 and all weapons do 1D6 damage (meaning that there is only one variable).

Just some thoughts.

[Matthew]

Duh! I just realised today that the doubling of horse to foot combat probabilities [i.e. dice] is probably the root of why lances do double damage in AD&D!