So, all of this math stuff started with me trying to build some 1k lists, and being frustrated that I kept running out of Troop choices. I noticed I wasn’t taking much by way of heroes or monsters.. so I thought to myself, well… what if I take a few Ballistae instead of the Troopsized infantry shooters I was trying to squeeze in? That’s save me some points, and free up some Troop slots.
I really liked my solution, and drew up a new MSU list I am very eager to test out and report back on… (hopefully in the next week). While my solution fixed some list concerns, I found myself wondering how how mathematically viable my solution was.
Like I said before, I am not a statistician. So I tried to break my question up into more manageable components, part one was figuring out what shooters I should hypothetically be running, with was <this post>.
I was surprised to find that an Arquebusier Horde is not only the deadliest ranged infantry unit, but also the most points efficient. In general Hordes are the best “bangforyourbuck,” but most troop choices aren’t too far behind, and you’re paying for troops to increase your list’s versatility (more drops) or ingame options (splitting fire among enemy units/avoiding overkill). So, after all that math, I had my goto unit to compare against the Ballista: the Arquebusier Troop.
So…. Arquebusier Troop vs Ballista!
I had no idea how to compare the two, since the Ballista fires one shot, and if it hits, turns into multiple hits. Perhaps my approach is totally wrong, but I decided to break it all down to a single attack to start.
Both have a 5+ to hit, or a .33 chance to hit with each volley. The Arquebisuer has Pierce 2; the Ballisa Pierce 3, so assuming a hit, their chance to wound for a few Defense stats is below.
De3

De4

De5

De6
 
Arq. Troop

0.83

0.83

0.67

0.50

Ballista

0.83

0.83

0.83

0.67

Again, this may be wrong approach, but I decided to look at the chance each attack would have to wound, taking the (.33 chance to hit with a shot) multiplied by (a single hit’s chance to wound, shown above). That gives me the chance of each shot to hit and wound (I think):
De3

De4

De5

De6
 
Arq. Troop

0.28

0.28

0.22

0.17

Ballista

0.28

0.28

0.28

0.22

That looks right to me. The impediment is hitting, but provided that the shot hits, the solid Pierce values help insure that it wounds. The pseudo plateau also makes since, since Pierce eventually gets the roll to one or below, but rolls of 1 are always failing.
Alright, so that looks like the math for each shot. But the hard stuff is still to come… Units have multiple shots per volley… and the bonkerz D3+2 Ballista is still wrecking my noggin.
The unit stuff should be easier? I think? We take their communal shots, and multiply that number versus our values, to get the average damage per volley. That comes out to what I came up with previously, so uh… I think that’s good? Probably wasted some time here.
De3

De4

De5

De6
 
Arq. Troop

2.22

2.22

1.78

1.33

The Ballista is harder… it has one shot, but Blast means a connecting shot will result in D3+2 hits against the unit. If we take the chance to hit for a shot, and then assume 4 hits (the average roll for D3+2), we get the following average wounds per volley.
De3

De4

De5

De6
 
Ballistae

1.1

1.1

1.1

0.9

Fortunately, at this point, my googlefu turned up this thread after a few hours spent looking at various probability tutorials trying to figure out how to proofread my approaches. I was happy to see Blast was messing with people far more math savvy than I!
As discussed in the thread, the math is a little off, as this is looking at an average, not probability. Two thirds of the time, the Ballista will miss and do nothing. Averaging out volleys shot and damage dealt would get the values above.
So, it’s imprecise. But, it is largely comparable to the values I was using before. Since we were looking to compare the two units, that’s a win.
So, Like before, let’s take the points of each unit, and divide it by the number of average damage dealt, and see what unit will, on average, give us the best efficiency. Again, a lower number is more desirable, as you are spending fewer points to deal the damage.
De3

De4

De5

De6
 
Ballista

54.55

54.55

54.55

66.67

Arq. Troop

45.00

45.00

56.25

75.00

Man, these Arquebusiers! The Arquebusier Troops coming out on top against Def3 and Def4 shouldn’t be a surprise, given our previous testing and some mental math (the Arquebusiers being on average about twice as deadly as the Ballista for less than twice the points). Also to be expected is a little “overkill” factor, as any additional Pierce you have after getting the damage roll down to a 2+ is just wasted points.
So, the Ballista really only comes into its own mathematically once that extra Pierce can kick in, which happens at Def5. Still, that is a tiny difference.
I still want to experiment with a few Ballistae in my MSU lists to free up Troop slots… but man, it sounds like for someone just starting their army, Arquebusiers are a good buy.