Ballistic Theory - the Accuracy of Ordnance

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+++Thought For The Day: The Summary Execution is the most accurate attack in 40K.+++

Ordnance is a key part of any Imperial Guard army. Whilst Heavy Bolter teams will slay hordes of Orks, and Lascannons defile enemy armour, a good Guard officer knows that Battle Cannons level everything.

There are some, however, who question the accuracy of these weapons [and are duly executed for their crimes]. For this reason, we have compiled ballistic sampling of Ordnance, and calculated the efficiency of the weapon as accurately as can be reasonably expected.

Adepta Hospitaller Health Warning:
This article contains a system of formulas commonly known as "Mathhammer". Mathhammer requires a level of free thinking deemed unfitting for a true, loyal Guardsmen; ignorance is your shield! To trust solely in this blasphemous 'Mathematics' is to condemn yourself to defeat! Trust in the Emperor, and His Righteous Fury shall guide your Ordnance straight and true to the target! Remember; an open mind is a Heretical mind!

To establish a system by which the true ballistic accuracy of Ordnance can be compared, we will calculate the accuracy based on previous Editions. We will also use four common terms, each of which assuming we are firing at a squad of standard-based Infantry, with a central model under the hole:

Direct Hit: This represents a shot with 0" scatter, or a 'Hit' roll on the dice.
Minor Drift: Minor Drift is a shot that scatters, but not enough that the initial target under the whole is missed by the template.
Major Drift: Major Drift is a shot that has missed the target under the hole, but could still have reasonably hit other members of the unit.
Miss: The Miss is where there is no acceptable level of probability that the initial target was struck at all by the shot.

3rd / 4th Edition:
First, let us define the mechanic; the shot rolls a Scatter Dice, and D6 for the number of inches scattered. Moving vehicles in 4th rolled 2D6 and picked the highest, but as the average roll of 3.5" per dice is unchanged, this is a moot point.

First, the Scatter Dice itself gives a 1/3rd chance of a Direct Hit.
Assuming a scatter, there is a 1/3rd chance of a Minor Drift, a 1/2 chance of a Major Drift (assuming equal spacing around the central model), and a 1/6th chance of a total loss.

However, 3rd and 4th factored in 'Partial Hits', meaning there was a 50% chance a model only partially under the template was hit... thus, our accuracy for the Major Drift is significantly lower; easily 1/6th less efficient.

Taking that into account, we have:

33% chance of a Direct Hit.
22% chance of Minor Drift.
22% chance of Major Drift.
22% chance of a Miss.

5th Edition:
5th Edition is somewhat more complex, in that the Scatter distance is theoretically higher, but modified by Ballistic Skill.

As we use D6+D6, we have a total of 36 possible roll combinations.

Of these, due to the -3 modifier, 1/12 will result in a 0" scatter - a Direct Hit.
7/36 will result in a scatter of 1-2", resulting in a Minor Drift.
16/36 will result in a scatter of 3-5", resulting in a Major Drift.
The remaining 10/36 represent a Miss.

So, what does that mean mathematically?

Given we still have a 1/3rd chance of rolling a Hit on the scatter dice, the odds are:

39% chance of a Direct Hit.
13% chance of a Minor Drift.
30% chance of a Major Drift.
18% chance of a Miss.

As you can see, the chance in Editions has done something rather interesting for our Ordnance; it's made us more accurate in terms of direct hits, less accurate in terms of landing the shot "there or there abouts", but significantly more accurate if you are just trying to hit the broad side of a barn.

The somewhat worrying conclusion, however, is that when you combine Direct Hits and Minor Drifts, the end result is 3% lower in 5th; this means that, whilst we are better at hitting the squad as a whole, we are in fact less accurate at singling out a specific model.

Operational Revision - How To Employ Ordnance:
The maths proves it; statistically, you are more likely to land a direct hit now than you were before. However, when you take into account Scatter rolls, you are less able to strike a specific target. That said, Ordnance cannot 'single out' a model in the unit, so there is no tactical advantage to this behaviour.

My conclusion, therefore, would be to use Ordnance primarily to take out large formations. However, the odds are better now that you can hit smaller units, or individual models, as we score more Direct Hits and the "Partials" rule does not apply anymore.

Limitations of the Mathhammer:
Mathhammer has two notable limitations. The first is that it cannot accurately calculate how many you will hit; most squads are not spread out evenly in a circular pattern. It is entirely possible, even probable, that a 3-4" scatter could leave you with no models under the template. The Mathhammer has assumed that, even if you scatter 5" (the diameter of the Blast Marker), there will still be at least one model of the target squad under it.

The 3rd / 4th Edition rules also do not accurately take into account Partials in the Minor Drift; a Scatter Roll of 3" would result in the initial model only being hit on a 4+, further complicating the maths.

Tank-hunting officers will also note that, in 4th and 5th, you need the target under the hole of the Template to do any real damage... a fact that did not exist in 3rd. Whilst I've not run the math, it's safe to say that 3rd Edition was when the Ordnance was king for tank-hunting; it has since lost that crown.

Just remember; like the Sisters Hospitaller said, Faith is your best weapon. If you play every game with a calculator and a checklist of the probable outcomes, you aren't going to win games. If you're willing to say "damn the odds, load HE shells and fire at that Chaos Predator!", then the Emperor may well smile on you and, in spite of statistical probability, blast your target into a smoking heap of scrap metal.

Mathhammer is not the be all and end all, but it is something to keep in mind; in terms of hitting exactly where you want to, you've got more chance of pulling it off with a Lasgun than a Battle Cannon.

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