Large Gauss vs. Small Gauss

[HartLord (OP)]

I've looked around awhile and found nothing on this: the efficiency of normal sized gauss turrets vs the smallest possible sized gauss turrets.

My question is: are size 30 quad gauss turrets more efficient than ten times their number in size 3 quad gauss turrets against large numbers of hard-to-hit missiles?

[telegraph]

numbers: one turret - one salvo. the more turrets you have - the more salvos you will be able to engage (provided you have enough firecontrol.)

chances: if you have one missile - one full-size gun with 100% hit chance would be better then 10x 10% to-hit guns. 1/(1 - 0.9^10) times better. That is without traking bonus and crew grade and other factors.

[HartLord (OP)]

Actually the accuracy is even worse than that at about 8%, but what I'm hoping is that some of the number savvy people around here will take notice and work out whether or not the sheer amount of dakka being sent out by all the tiny turrets you can cram on a ship will make up the difference.

[Nightstar]

For gauss, accuracy * size divisor is a constant 1. More less accurate guns will have a different probability spread that's generally worse. However. With crew/officer bonuses, it's possible to get >100% accuracy on large guns. In that particular situation, smaller guns are better.

[Hawkeye]

but what I'm hoping is that some of the number savvy people around here will take notice and work out whether or not the sheer amount of dakka being sent out by all the tiny turrets you can cram on a ship will make up the difference.

Oh no!!!! What have you done?
Now we will be drowned in pages and pages of math formulas!!

We are doomed, doomed I say!                

[solidcordon]

If the missile salvo passes through your engagement range during a 5 second phase then. . .

Final fire Point defense at 10k km will probably be best.

If your smallest turrets have an 8% chance to hit you'll need 12. 5 shots to be 100% sure of a hit, on average.

More guns per turret means more chances to hit each salvo, higher rate of fire means more shots per phase per salvo (I think)
Unless you can improve your turret tracking gear or fire control tracking speed then I'd recommend as much dakka from smaller turrets as you can get.  Alternatively use rapid fire size 1 missile launchers and anti-missiles.

Most missiles I have seen will pass through the engagement range of the largest gauss cannon available in 5 seconds, for multi layered point defense it may be worth putting some laser turrets on but even then your tracking gear and fire control is going to let you down.

[HartLord (OP)]

I'd like to keep this contained to small gauss cannons vs large gauss cannons.

I started on this because I didn't feel like managing AMMs at the time, and large gauss turrets seemed like they would be easily overwhelmed by a large number of small missiles.  From one engagement against prespoilers I found that most of their 20 or so AMMs each wave would be shot down by the high number of turrets, but that's the only time I've used them in combat.

[telegraph]

If your smallest turrets have an 8% chance to hit you'll need 12. 5 shots to be 100% sure of a hit, on average.

Probability does not work that way. for N 8% shots (because turrets may have different number of barrels and barrels may do different number of shots.) probability to hit a single missile(because salvos may have different number of missiles) would be 1 - 0.92^N. It will never-ever be 100%.
On the other hand you might be ok with 99.9% chance, in which case there might be some situations where quantity is preferable to quality. The ratio for each situation (salvo size, missile payload, hit chances based on speed, number of salvos, intended crew quality, ship type and commnder boni (for those fighters)) should be calculated separately, and can get quite complex at times. Those calculations are generally not needed though.

As a rule of a thumb you can use
1. half-sized gauss cannons on small ships that travel in groups to allow for traking and crew boni.
2. full-sized gauss cannons on designated area-defence ships or large battlecruisers
3. smaller gauss cannons on fighters because you are really hard pressed on space and commander bonus is really important there.

[sublight]

Math Incoming!!
(this wall of math is short, I promise)

The numbers are balanced so that any given HS worth of Gauss cannons will have about the same average number of hits.

A 6 HS cannon has 100% chance of one hit, modified by accuracy, for an average of one hit per firing tech level.

The smallest 0.5 HS cannon has an 8% accuracy modifier, and one could fit 12x of those in for a single full-sized one.

P(x) = probability of X hits = nCr(12,x) * 0.08^x * 0.92^(12-x)
P(0) = 36.8%
P(1) = 38.4%
P(2) = 18.3%
P(3) = 5.3%
P(4) = 1.0%
P(5+) = 0.2%
Average hits = Sum(P(x) * x) = 0.96

As you can see, the tinniest gauss cannon takes an increased chance of everything missing in return for greater flexibility in weapon assignments plus a chance for additional hits.

Note: turrets give a 10% size reduction for a quad over a single. This lets us squeeze a 13th 0.5 hs turret into the same space as a single full sized 6HS version.
13x 0.5 gauss cannons have a hit average of ~1.04.

I'd recommend only using reduced size if you lack the room for a triple or quad turret, and even then to not go below 1HS to avoid the frailty of a 0-HTK components.

For reference nCr(N,x) is the number of combinations possible when selecting x items out of N possible, where order does not matter.
nCr(N,x) = N!/(x!*(N-x)!)

[telegraph]

I'd recommend only using reduced size if you lack the room for a triple or quad turret, and even then to not go below 1HS to avoid the frailty of a 0-HTK components.

Only if you are not going to utilize additional boni. long traking and high quality crew (not even mentioning a jedi-class fighter pilot) could drive to-hit chance of a full-sized cannon well over 100%, so using those on a non-critical vessels would be a waste.

Edit:
Also you calculation example is correct for RoF 1 gauss cannons only. If you take another extreme edge of RoF 5 cannons - the battary of 12 8% barrels will have a 99.3% chance of scoring a hit (60 shots).

For OP:
There could be a situation when enemy is bombarding you with a large number of single-missile salvos (I would be such an enemy) 4 shots out of 5 would be wasted for every full-sized gauss cannon. Smaller barrels may be better distributed to cover more salvos:

Let's assume your ship is attacked by 10 1-missile salvos.
You have 30HS space on the ship for defensive RoF 5 gauss cannons. (I will ignore FC, turreting and any additional boni here for simplicity)

If you decided to use five full-sized cannons - you would destroy 5 missiles, and take a hit from 5 others. so it is a 50% protection.

If you decided to use 10 half-sized cannons - you would engage every salvo, but you chance of hitting any particular missile would be 1-0.5^5 = 96.9%. The chance for you hitting all of them (which was quite impossible in former full-sized setup) would be 0.969^10 = 72.8%

You can now calculate the average damage in the second setup using the formula given by sublight.

That is all just to prove that optimal setup will differ from greatly your situation.

[Nightstar]

Yeah, a fire control per half size gauss cannon isn't remotely realistic. Turret FCs are EXPENSIVE.

You also want to calculate the chance for five hits (same as full size) from the small RoF 5 cannons.

Half size make sense if you're likely to break 100% hit rate. Current fighter combat bonuses make massed 0.5-0.6 HS cannons good on fighters. Otherwise, smaller are slightly worse in exchange for lower research costs.

[Paul M]

Just a suggestion:

Before worrying about the math define the situation.  What are you comparing to what...what are you trying to stop...what technology do you want to use.  At that point you can do a sensible analysis.  Also keep in mind the analysis is only valid for the conditions you define, it may be valid for other conditions but you can't be sure.  But without a proper starting point your math goes no where. 

Otherwise the technical term that applies to the analysis is: Garbage In, Garbage Out (GIGO).

[HartLord (OP)]

Thanks sublight for that particularly helpful post.

And that wasn't a wall of math! You lied to me.

[Paul M]

What sublight put up is also the binomial distribution, which is the correct distribution to use to solve these sorts of questions.

[Jorgen_CAB]

My take are pretty much to reduce the size of the Gauss cannon with each advance in number of shots to keep each turret downing approximately the same number of missiles per salvo.

At 3 shots per barrel I generally have half size cannon in quad mounts with a dedicated fire-control. Should be enough to engage fighter launched missiles salvos in combination with rudimentary AMM coverage.

You can also in some rare cases make smaller guns more efficient as said above with crew grade increasing the to hit rate. Unfortunately the tracking bonus tech don't seem to work correctly currently, but if it did it could add even further use to smaller Gauss cannons.