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PostPosted: Sat Nov 10, 2012 11:06 pm 
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Here is a list of expected (average) number of wounds from a hit with a bolt thrower. The values are given for models with varying armor saves, toughness scores, number of wounds. The values can be compared between volley and bolt firing option. They take into account penetration through multiple ranks. All values assume you have already hit (with one bolt or all 6 shots from the volley) and don't take into account aspects like ward or regen saves.

Results:
As expected, we see more wounds being done to heavily armored troops by the bolt, and to lightly armored troops by the volley. For monsters (many wounds, no penetration) a volley is always the optimal choice when no armor is involved.
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Khorne Juggernaughts, chaos knights, demigryphs and empire knights all suffer more wounds from the bolt option when a shot is being fired down their flank (or through several ranks). Ogres suffer slightly more wounds from the bolt option. Volleys outperformed bolts against monsters of any toughness, except at toughness 6 where there is no difference. With a 4+ armor save bolts become favorable against T6 monsters (aracknarok etc) while volleys remain best against T5 monsters (hydras etc) or monsters with toughness below 5 or above 6.

The results have been updated to include figures for Brettonian Knights. It turns out you'd best hit them with bolts as well, even in the face of their 5+ ward saves. This even applies for toughness values from 1-7, covering everything from withering to stoneskin + throne of vines. It's not even worth nerfing their toughness if you're going to be shooting bolt throwers at them. The bolt is still more effective than the volley and does not really improve against lowered toughness.

Discussion:

The intuition that you should use a bolt against highly armored targets when firing from the flank is supported by these findings; almost all of the common knightly threats are better dealt with using a bolt when you have the opportunity to penetrate through multiple foes. When you are not firing through ranks (at least 3) of heavily armored foes, the volley option is almost always superior.

One interesting trend is that for a bolt penetrating ranks, high toughness targets suffer more damage as their number of wounds increases, but low toughness targets suffer fewer wounds. This occurs because most of the wounds inflicted on low toughness models are a result of penetrating ranks, but increasing the number of wounds per model greatly decreases the chance of rank penetration occurring. Against high toughness models, most of the wounds inflicted come from the front rank, and increasing the number of wounds per model allows the bolt to take advantage of the multiple wounds rule.

These results assume that bolts fired at units are able to strike up to 6 targets by penetrating ranks. However, the assumption of the 6th, 5th or even 4th target adds very few wounds to the average wounds inflicted value, because even against low toughness foes the chance of penetrating so many ranks is very slim. For a shot fired down the flank of demigryphs, chaos knights, etc these results are a fairly accurate approximation.

I invite the harshest criticism and will update this if anyone can think of a way to make it better. I can also provide the macro-enabled excel sheet with the formulas used to create this.

Critiques:
Ward save issue, noticed by me: ward saves and regen can have a significant effect on how a bolt performs vs a volley. When you're shooting at a monster or a lone character, ward saves affect the average number of wounds done from each source by the same ration. For example, a HPA will halve the number of expected wounds from a volley or bolt thanks to its 4+ regen. This does no apply to penetrating ranks however. A bolt fired at a unit with a 5+ ward save will wound the first target hit (2/3) times as often, the second (2/3)^2 times as often, the third (2/3)^3 times as often, etc. This reduces the effectiveness of a bolt disproportionately more than it does the effect of a volley, and could be highly relevant against brettonian cavalry. I will add a calculation for brettonian cavalry shortly, and consider adding stats for more ward save scenarios.

How many kills, notice by Swordmaster of Hoeth: What I really want to know when I fire a bolt at a unit is, how many guys can I expect to kill? What are the chances of killing 1, 2, 3 etc... I have updated the figures to show this relevant information.

Updates: I have now included some figures for Brettonian knights. I might update this to include other common ward/regen sources, such as trolls, ironcurse icon, earthblood.
Nov 12, 2012: Fixed some math errors for bolts, added numbers for bolts against targets with regen or ward saves

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Last edited by Eirik on Mon Nov 12, 2012 11:15 pm, edited 5 times in total.

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PostPosted: Sun Nov 11, 2012 5:43 am 
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Hi eriktheguy,

Thanks for the very interesting results. I am sure many players will appreciate the effort you put behind the calculations (no matter the method).

In terms of general approach I think, however, that average number of wounds to be expected to be inflicted is more difficult and less conclusive to use as a guideline and help in decision making. The main reason for that is the fact we deal with fractions. If you only put a number like 2.11 in the table then you can of course claim that you can expect 2 wounds to be inflicted. But then it is quite vague if you compare it to 2.25 because obviously it is better result but should I interpret that as enough chance to get 3 wounds instead?

Also, in calculations like that it is good to have either error bar or confidence interval. Then it gives you better picture of what kind of results are to be expected.

Personally, I prefer different approach. Also because we do not have fractions when rolling dice. I would rather know what is the probability to penetrate 1, 2 or 3 ranks of a target with particular toughness for example. Or what is a probability to inflict particular number of wounds with a volley under specific circumstances. I think such approach is clearer and helps in decision making.

Having said that I wanted to thank you once more for nice job.

Cheers!

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PostPosted: Sun Nov 11, 2012 3:37 pm 
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I think a useful statistic along those lines would be the probability of getting at least X kills. I could add that as a dimension for the bolt lines, because they don't have an armor dimension to them.

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PostPosted: Sun Nov 11, 2012 4:58 pm 
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I think trying to generalise to this point though sometimes misses other important factors. Ward saves can seriously damage the effectiveness of a single bolt. For example against bretonnian players

Their knights get a 6+ ward save against S4 or less and a 5+ ward save against S5 or more. Supposing there is 12 of them....

Suppose we hit them with a single bolt at short range (up to 24 inches)... =(2/3)*(5/6)*(2/3)(1+(5/6)*(2/3)(1+(2/3)*(5/6)*(1+(1/2)*(5/6))))... excel isnt working on my laptop please could someone plug this in for me?

Whereas with the multiple shots it is =6*(2/3)*(2/3)*(1/2)*(5/6)

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PostPosted: Sun Nov 11, 2012 6:15 pm 
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Hey andrew, your point about the ward saves is very important. Ward saves reduce the wounds from a volley by a scalar factor, and from a bolt by a higher amount. The chance to wound the nth rank is reduced by the nth power of the ward save. For example, a 5+ ward means you're 2/3 as likely to affect the 1st rank, 4/9 as likely to affect the 2nd, etc. I sort of ninja's you on this when I added it to my OP in an edit a bit ago.

I'm going to update the excel with this math.

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PostPosted: Sun Nov 11, 2012 6:57 pm 
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I have updated to include chance to penetrate up to 3 ranks.

I have updated to include figures for Brettonian knights, who are an attractive and interesting target for bolt throwers.

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PostPosted: Mon Nov 12, 2012 9:16 am 
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Andrew_uk wrote:
Suppose we hit them with a single bolt at short range (up to 24 inches)... =(2/3)*(5/6)*(2/3)(1+(5/6)*(2/3)(1+(2/3)*(5/6)*(1+(1/2)*(5/6))))... excel isnt working on my laptop please could someone plug this in for me?

Whereas with the multiple shots it is =6*(2/3)*(2/3)*(1/2)*(5/6)

0.74 for the first and 1.11 for the second.

I almost always have excel lying around...

Rod

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PostPosted: Mon Nov 12, 2012 5:55 pm 
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Cool :) so then against bretts rather counter intuitively it is best to use the multiple shots!

Even a 5+ ward can ruin the efectiveness of a bolt thrower

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PostPosted: Mon Nov 12, 2012 11:16 pm 
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Updated with some corrections, volley IS better against knights than a bolt. Oops.
Added figures for bolt firing through ward and regen saves.

Will soon update with chance to do at least X wounds with volley.

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PostPosted: Mon Nov 12, 2012 11:33 pm 
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Your analysis is interesting. Personally though when I Mathhammer things like this I don't like to make assumptions, such as the bolt always hitting and all 6 volley shots always hitting. I know it's more complicated (since you would need to take a variety of possible to hit values into account too) but you'll find some interesting results. Overall though, as should be intuitively thought, you'll see taking average to hit values into account will show you that the bolt has usually higher damage potential but is less reliable (either hit or miss, more likely to wound, no armor) compared to the volley which will have a smaller range of damage potential, but will be more reliable within that range (on average a few shots will always hit, less likely to wound, less armor pen).

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PostPosted: Tue Nov 13, 2012 1:42 am 
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You can modify any value in this sheet to the true average by multiplying it by your to-hit chance. If you multiply the 'average wounds' value for volley or bolt by your to-hit chance, you get the new true average that takes to hit into account. You also get the true chance of killing at least X models with a bolt if you multiply by the two-hit chance.

When I add the table which indicates the chance to 'inflict at least x wounds' for volley fire, I will need to relax this assumption. I will probably produce charts for chance to do at least X wounds with a volley for hitting on 3s, 4s and 5s. The actual value I post on the table will not be the true calculated value, but will be multiplied by a factor of 1/(hit chance) so that it can be directly compared to the same values for the bolt charts.
For example, if a volley fired at 4+ to hit has a 10% chance to do at least 1 wound to a given unit, I will report the value as 20%. Then you can look at the chance for a bolt to do at least 1w to the same unit. If the value is over 20% the bolt is better, if it is under 20% the volley is better.

TL/DR: You can compare the average # of wounds between bolt and volley without worrying about the hit chance. You need to take hit chance into consideration when considering chance to do at least X wounds.

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PostPosted: Tue Nov 13, 2012 6:49 pm 
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I agree. My point was that I wasn't just looking at the average, I was also looking at the probable distribution of the results. The range and the chances within it naturally are different in the two firing modes. If I was to fire a RBT, I would personally find the information of what the likely range of results is to be as useful as what the most likely result is. In other words I like knowing the distribution in addition to the mean, as it were.

For example, I'm firing at a Toughness 6 Monster with 4+ Armor from extreme range and due to some other effects I'd be hitting at a 5+ (i.e. a risky shot). The average results assuming hits would merely say that the single shot is slightly better.

But if I do the full math:

Bolt: Chance to Hit * Chance to Wound * Chance to Bypass Armor * Chance to Bypass Ward
(2/6) * (3/6) * (1) * (1) = 16.7% chance to wound which distributes to:
0 Wounds: 83.3%
1 Wound: 5.6%
2 Wounds: 5.6%
3 Wounds 5.6%
(Yes rounding causes that to hit 100.1%) the true values are 83 1/3 and 5 5/9.)

Volley: Chance to Hit * Chance to Wound * Chance to Bypass Armor * Chance to Bypass Ward
(2/6) * (1/6) * (4/6) * (1) = 3.7% chance for each bolt to wound
So the distribution since there are 6 shots is more varried:
0 Wounds: 79.7%
1 Wound: 18.4%
2 Wounds: 1.7%
3 Wounds: 0.1%
4 Wounds: negligible
5 Wounds: negligible
6 Wounds: negligible

So reading this you can paint the picture that while yes, the bolt gets a higher expected number of wounds, the volley gives you a more consistent chance of scoring at least least 1 wound. The difference is slight in this case, but you'll get more interesting results when you play with various to hit values and targets. The bottomline is that the bolt is all or nothing, but 6 shots for lesser affect can offset d3 wounds (despite lower armor penetration and strength) under proper circumstances.

So while this isn't the best example to show that, I'd just point out what happens if the Monster in question has exactly one Wound left? Knowing this, which would you rather fire?

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PostPosted: Tue Nov 13, 2012 7:45 pm 
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In most cases the chance of a volley landing at least 1 wound will be better than that for the bolt. This is my plan for the next addition to the tables.

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