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TL Blast Mathhammer
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Old 16 Feb 2009, 06:43   #1 (permalink)
Shas'Vre
 
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Default TL Blast Mathhammer

I'm not entirely sure where this should be, as it doesn't ONLY deal with Tau weaponry (though I'm fairly sure Tau have the most of this type of weaponry, and I'm doing it with the Tau Fusion Cannon in mind)...

Also, for those do-it-yourselfers around, if you can't figure out how I calculated these, I'll PM 'em the Excel sheet to those that ask politely.



OK! So, some of you might be aware of Forgeworld's Imperial Armor III - Taros Campaign book. Amongst other things, this book includes several alternative Hammerhead turrets, most important to this post the Twin-Linked Fusion Cannons.

A most delightful weapon to imagine, this is a devastating weapon that uses a small blast AND is Twin-Linked, which means that after you roll the scatter dice when firing this weapon, you can choose to reroll the scatter die if you find the resulting location unsatisfactory (say, on top of your nearby Shas'o and his retinue...).

So what does mathhammer have to do with this, you ask? Well, I was wondering about the relative accuracy of this weapon when factoring in the potential for a Reroll. Now, sometimes one will be so incredibly lucky that a blast will scatter off the target and onto something just as wonderful to be covered (say, that Basilisk out of sight and range, rather than the Leman Russ whose weapon is destroyed already anyway), but counting on things like that is like counting on the double 1s when you've lost combat by 13 models or on double 6s when your opponent's Librarian is in Vortex of Doom range...

This math doesn't take that into account, and no math really can, though if the model IS mostly or entirely surrounded by models roughly X inches away, this math might be somewhat useful.

On to the numbers.

First considered is a TLed blast's scatter at BS 4. Notation-wise, when I say >= X" I'm referring to a reroll of all results that scatter X" or more away from the initial target. I'll be using approximate percentages (you really don't want me to have to deal with hundred-thousandths of a percent, do you?), but won't have the % sign. I'm not considering >= 6" or higher, as that's an unreasonable scatter even for a large ork mob, generally, and it's math I don't feel is worth doing, even with Excel's help.

Formatting sucks, but I apparently suck even harder with html table formatting, and this is hopefully at least readable.

scatter 0" 1" 2" 3" 4" 5" 6" 7" 8"
>= 1" = 69.1 4.1 5.1 6.2 5.1 4.1 3.1 2.1 1.0
>= 2" = 65.8 10.9 4.5 5.3 4.5 3.6 2.7 1.8 0.9
>= 3" = 61.7 10.3 12.9 4.3 3.6 2.9 2.2 1.4 0.7
>= 4" = 56.8 9.5 11.8 14.2 2.5 2.1 1.5 1.0 0.5
>= 5" = 52.7 8.8 11.0 13.2 11.0 1.4 1.0 0.7 0.3

Its a similar pattern for BS 5, a single ML boost away.

scatter 0" 1" 2" 3" 4" 5" 6" 7"
>= 1" = 76.8 4.5 5.3 4.5 3.6 2.7 1.8 0.9
>= 2" = 72.0 12.9 4.3 3.6 2.9 2.2 1.4 0.7
>= 3" = 66.3 11.8 14.2 2.6 2.1 1.5 1.0 0.5
>= 4" = 61.5 11.0 13.2 11.0 1.4 1.0 0.6 0.3
>= 5" = 57.6 10.3 12.3 10.3 8.2 0.6 0.4 0.2

Suffice to say that a similar pattern occurs for all data? BS 3 or less is obviously going to be less accurate, while BS 6 or higher more accurate.

Well, TL Blast weapons sure aren't as accurate as they used to be in 4th edition, that's for sure. Gah, how I wish I'd played this weapon when you rolled to hit like normal, then placed the blast template on the model you desired... and when cover wasn't as much of an issue.

Its pretty easy to see from the data that there is a tradeoff between reliability and accuracy...

... perhaps i should explain that.

Reliability, for this post, is the width of the range of results for a majority of the cases - if 1 result out of 20 possible results occurs 99% of the time, that's on the extreme side of reliable. If all 20 results out of 20 possible are nearly equally likely, that's an extremely unreliable weapon.

Accuracy, for this post, is the consistency of hitting a particular range of results. If 99% of results occur within the preferred range, that's extremely accurate. If only 1% of results occur within the preferred range, that's extremely inaccurate.

To show my point, consider the TL Fusion Cannon hunting one of its preferred targets, TEQs, who have just Deepstruck nearby...

Unless those TEQs ran in the shooting phase (and for this example, let's assume they had a couple Cyclone Missile Launchers they thought worth firing THAT round), they'd be all bunched up. When all bunched up, even if there's ten of 'em, scattering 2" or more away means that you're missing a majority of the unit, or missing that unit entirely. That said, a 1" scatter is still very potent - it might miss 2-4 of a 10-strong unit of terminators, or 2 of a 5-strong unit, but that's still getting a LOT of expensive TEQs underneath a very nasty blast.

So we have our preferred range, 0 to 1" scatter. Anything more, and barring extreme luck and a nearby enemy, its a wasted shot or potentially even dangerous to one's own units! To be considered accurate, we want as much of our results to scatter 1" or less as possible.

What happens if we decided to reroll all scatter dice that don't result in 0" scatter? Well, we see that roughly 69% of the time we'd hit spot-on, and 4% of the time we'd scatter only 1". That leaves us with 27% of the time where our shot is mostly or entirely wasted.

What happens if we reroll all scatter dice that don't result in 1" or less scatter?
Roughly 66% of the time we're right on, 11% of the time we're 1" scatter away. That leaves us with only 23% of the time giving us a wasted shot.

What about 2" or less scatter? ~62% right on and 10% 1" away... worse in every way than both other scenarios.

So clearly we have a choice to make - do we want the more reliable shot that hits EVERY TEQ by saying "0 inch scatter or bust!" or do we want the more accurate shot that HITS TEQS more often?

This is a choice that can be more clearly differentiated by going into the particular formation of the TEQs and probability of directional scatter... and in a lot of cases, the 1" scatter could easily end up being worth 1/2 as many TEQs under the blast as a 0" scatter. At that point, the reliability of the 0" scatter or bust choice seems to have greater weight.

To the same extent, except perhaps even more pronounced, one might see the same decision, made for the same reason, when targetting that Dreadnaught that just got Drop-podded nearby. 1" scatter might cause the hole to miss it entirely, and would only really be partially useful if it scatters towards its back armor.




Not all our targets are going to be bunched up or (relatively) small vehicles, however. Sometimes they'll spread out, perhaps to the maximum 2" coherency. At that point, the gun's potential drops, and its accuracy is based on a wider range of distances (made especially clear when a 1" scatter actually puts MORE models under the template than would occur if it hadn't scattered at all).

In those cases, one might prefer a looser rubric of 3" or 4" scatter or less.


Another thing to note is that, while slight, there's an understandable secondary pattern to the scatter die results - for BS 4, 3" scatter is, aside for 0" scatter, the most common distance to scatter for any given scatter roll. This is, of course, due to more 7 results occuring in 2d6 than any other result. To say, though, that the "average" scatter is 3" is a most horrendous mistake to make - even for blasts that are not TLed, a majority % of BS 4 blasts will scatter 1" or less.





I think that concludes this post - I hope, for those not at least instinctually aware of the consequences of TL Blasts and the way scatter works, that this post is useful in all your endeavors!

Edit: Unless you're firing TL blasts at MY army! :P :-*
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Old 16 Feb 2009, 15:48   #2 (permalink)
Shas'O
 
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Default Re: TL Blast Mathhammer

Ah, the missing % threw me off a bit.

Mind sharing some of your formulas?
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Old 16 Feb 2009, 17:44   #3 (permalink)
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Default Re: TL Blast Mathhammer

Why not.

Figuring out % of scatter for a single scatter die roll (say, for BS 4) is pretty simple:

For 0" roll, you add 1/3 (how often the "hit" mark comes up on the scatter dice itself) to 2/3 (how often the hit marker doesn't show) times 6/36 (how often 2, 3, and 4 result from 2d6).
For each inch scatter, you multiply 2/3 times X/36, where X is the number of potential die results for 2d6 that would cause that number of inches to scatter.

in notation, that'd be:
1/3 + (2/3)*(X/36)


To figure out Twinlinked accuracy, I had to force it into a category of choice for the first result - if its equal to or over X inches, reroll.

If I wanted to know what happens if you reroll all dice >= X", I created on excel a separate column, which had the probability of the initial result I was asking for (say, less than 1" scatter = .44), which I'll call P from now on.

For 0", I calculated how likely it was to get the 0" result on the first try (same 1/3 + 2/3 * X/36) then added (1 - P) times the same chance of getting 0" result on the second try.

I did this because, for the initial roll, the result of a 0" scatter would be kept - its within my rubrik of success. For all rolls outside of success (and when success occurs P times out of the total, 1 - P gives you the rest of results), I would have a chance of rolling a 0" scatter on the second dice, thus adding (1-P) * (1/3 + [2/3 * X/36]) to the initial chance of success.

For the probability of scatter distances outside of my preferred distance, the calculation is easier - it doesn't matter what my initial chance is of getting that distance, as that possibility is already accounted for in (1-P), so the calculation is merely (1-P) times the possibility of getting that distance on a scatter die roll, that is (2/3) * (X/36).

Nothing more complicated than that.

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Old 16 Feb 2009, 18:20   #4 (permalink)
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Default Re: TL Blast Mathhammer

Oh yea. Nothing more complicated than that. :P Maybe if I were still in Prob and Stats where this stuff was top of my mind daily instead of now when the most math I need is basic prob and stats and nothing more than fractions. :P

There used to be a mathhammer calc around here somewhere but it disappeared.
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