Scatter in 40k

There was once a young gunner from Cadia,
Who fired a missile over the top of his barrier-
It was aimed at some Orks,
But scattered off course,
And hit the Sarge in a very delicate area!

We know, more or less, how scatter works in 40k. But I've been interested in teasing out some of the mathematical detail, beyond the simple probability of scatter distance, to calculate the probability of a model at any given distance being hit by a shot that is subject to scatter.

Why? Well, two reasons, really. First, I want to look at the types of squad formations, and how a squad might best balance blast marker defence without diluting its onfield presence to unacceptable, or unworkable levels. Secondly, I think it would be useful to develop some kind of metric to math-hammer Blast weapons; ie, just how many shots is that 'Blast' designation worth, and will it be worth the points we pay for it?

So, before I present some common formations, and the expected number of hits to various weapons, here is a graphical representation of some of the data I have derived, for four of the most common cases. Probability is coded by colour, according to the scale on the right hand side of each pic (scale is the same for all pics). Distance from the epicentre is also labelled.

For kicks, try printing an image at 1:1 scale, and put your miniatures on in your favourite formation, against a typical opponent, and see how many of your guys are in the danger zone! (Actually, the resolution probably isn't good enough for that. Sorry! )

Small blast, BS4

Small blast, BS3

Small blast, BS 2

Small blast, BS 0

Of note, of course the central area diminishes in probability in direct proportion to BS, but also the outlying region that is most likely to be hit gradually shifts away from the centre as the average on 2d6 - BS increases. This reaches a head at BS0 (indirect fire without line of sight), where models 2 - 4" from the target are actually safer than models 5 - 8" away!

I've looked at the large blast marker too.

Large blast, BS 0

Large blast, BS 2

Large blast, BS 3

Large blast, BS 4

How did I calculate these, are you asking? I should point out that these all assume base size of 25mm, so standard infantry. I intend to do some other cases, like 40mm, and common vehicles, but it may take a little while - this has been cooking for some time.

Oh yeah, for method, I basically combined the probability of given scatter results with some trigonometry to derive the likelihood of one circle (the blast in question) overlapping another circle (the base of the model at any given point). I worked it out in intervals of 0.1", which while not perfect, I think will give sufficient granularity for our purpose.

To go through the entire mathematical rationale is probably a post in and of itself, so I'll only bother if people are interested. Otherwise, enjoy the charts, and I will attempt to get some data up on average hits on various squad sizes and formations soon!