Gunnery and dice probability

[Phthisis]

I see the word 'average' tossed around here all the time with reference to the expected number of hits a particular battery of weapons will cause.  For example, a player gets to roll 6 dice to hit a target with 5+ armor.  Based on this situation, most if not everyone, around here would say that such a shot would average 2 hits.  Eg..  There are 6 dice, and a 1/3 chance of both dice hitting, therefore an average roll would be 6x(1/3)=2.  This average is called a 'mean' average and, although easily and quickly calculated, it's somewhat misleading. First of all, results of dice rolls that have odds of less than 50% (like 1/3) do not follow an even distribution and are skewed towards failure.  Each individual dice has a greater chance of failure than success and that doesn't change if you roll 2 dice or 12.  Averages based off of means can be misleading this way.  It's like if you averaged the amount of money 4 men have when 3 of them only have $1 and the 4th has $100.  You'll get a skewed view of reality.  Mean averages aren't the best way to figure your odds when we are dealing with dice.  Using them to calculate the expected damage output of weapons batteries tends to give the player a feeling that their WBs never live up to expectations.  Second, the mean average doesn't really give you enough info to make a good tactical decision.  For example, what are the chances of missing completely?  What are the chances that I will roll more than 2 hits?  Knowing these probabilities can give you the tools you need to use your fleet effectively.


3 dice 5+ armor
0 hits = 29.63%
1 hits = 44.44%
2 hits = 22.21%
3 hits = 3.7%

4 dice 5+ armor
0 hits = 19.75%
1 hits = 39.5%
2 hits = 29.62%
3 hits = 9.87%
4 hits = 1.23%

5 dice 5+ armor
0 hits = 13.17%
1 hits = 32.92%
2 hits = 32.92%
3 hits = 16.45%
4 hits = 4.11%
5 hits = .41%

6 dice 5+ armor
0 hits = 8.78%
1 hits = 26.34%
2 hits = 32.92%
3 hits = 21.94%
4 hits = 8.22%
5 hits = 1.64%
6 hits = .13%

This is about all I have time for at the moment, but I can provide probabilities by request for any number of dice and armor value.


One thing to notice about weapons batteries is that the number of average hits doesn't go up proportionally to the number of dice that you add.  This is even more true when facing 6+ armor.  This is why many players say that their WBs never seem to perform up to their expectations.  For example, with 6WBs the mean average is 2 hits.  The likelihood of rolling 2 or fewer hits is about 68%.  The mean average of 2 hits seems to suggest otherwise because without understanding the actual probability, we tend to assume that the distribution of probability is even.

I believe that this goes some of the distance in explaining why some players prefer lances to WBs (or Murders with lances to Carnages).  As they always hit on a 4+ they tend to be much more reliable than low numbers of WB dice and also a lot more predictable.

[Taggerung]

It's really not that hard to calculate, and I am surprised more people don't do it.

for hitting on a 5+, it's a 1/3 for each dice rolled...so

(1/3) * (1/3) * (1/3) * (1/3) * (1/3) * (1/3) = all six dice hitting...just reduce or add the amount you are multiplying by to get your actual probability.

For calculating misses, you use (2/3) instead. In all cases, it's a 33% chance that each dice hits, independently of each other. So rolling 6 dice does not automatically = 2 hits. It's more likely that you will get a couple of hits, but that's more averages than statistics

[horizon]

Okay then. I want dice vs 4+ armour (eg lance shots) and hit chances.

thanks in advance.

So far nothing new.

And from what I read across the net new players often prefer lances. (I did) As you get more hang of tactics weapon batteries become cooler and cooler.


And I still fail to see how a lance murder outperforms a Carnage in a broadside.
4 wb + 2l vs 16wb.
Some averages for that?

[Taggerung]

So it's a (1/2) * (1/2) * how many more dice you are rolling.

[horizon]

I'm lazy, make me a list like in post 1.

[Phthisis]

It's not technically statistics either way.  This is probability.  The usual way is just averaging, which passes for neither statistics nor probability in the scientific world.

The method you describe is to calculate the chance of either hitting with everything or missing with everything.  Its everything in the middle thats hard to calculate and thats where most people get tripped up.  Arguing with Sigiroth made me realize that people are taking mean averages too literally.  But the actual likleihood of damage falls behind the mean average with each dice added.
I think its particularly interesting that whether rolling 6 dice or 3 dice, the probability is still heavily grouped around only 1 or 2 hits, and with 6 dice the odds of getting 5 or 6 gits is laughably low.  Mean averaging doesn't suggest this and I don't think most players have realistic expectations of how their gunnery will perform in game.

@Horizon
Glad you asked!

The Carnage utilizes the 3 dice profile above when abeam at 45, and the 6 dice profile when inside 30 in clean space.
Here's for the lance murder for both 45 and 30 bands:
2 dice 4+, 1 dice at 5+
0 hits = 16.66%
1 hits = 41.66%
2 hits = 33.33%
3 hits = 8.33%

Analysis: At 30-45 a clear advantage for the Murder.  Less likely to miss and much more likely to score hits at every level.
At 15-30, the Carnage clearly surpasses the Murder, which is to be expected, but not as much as one may have thought.  There is only about a 30% chance the Carnage will actually perform better than the lance Murder.  That is, unless there is a blast marker in the way.  Then the Murder would perform better again.  So I guess analysis is only for the first Carnage.

My main point about preference for the murder is that I prefer to fire broadsides from long range and I stay there throughout the game.  As long as I stay in that band, the lance Murder will do much better.  Combined with the opportunistic lance fire from the prow, I prefer the lance Murder.  I still take the Carnage, but building a gunline I'd favor the Murder.  If I wanted to get in close I'd take the Slaughter instead of the Carnage.

[horizon]

Hi,

where to start?

Ah yeah: staying at long range with Murder is not an option. Using the prow lances @ 60cm is closing with enemy, thus cannot stay at range. Thus it comes within 45cm. This is medium range.
You do know the broadsides are 45cm on the lance murder?

The Carnage can always be abeam and stay at 60cm, lend its 10wb to the long range lances of Desolator, Styx, Dev's, etc.

Also: your averages are vs abeam targets. Since the Murder is closing in a duel (since it needs to, the Carnage doesn't) with the Carnage, the Carnage gets more dice.

Carnage above 30cm:
-- vs closing : 8 dice wb
-- vs moving away : 6 dice wb
-- vs abeam : 3 dice wb

Murder above 30cm:
-- vs closing : 2 dice wb + 2 l
-- vs moving away : 1 dice wb + 2 l
-- vs abeam : 1 dice wb + 2 l

Under 30cm the differences goes up high.

And above 45cm the Carnage vs Murder = 10wb vs 2 l

It seems you isolate the single point the lance murder could be better (vs abeam) but forget the multiple benefits of the Carnage.

Look the murder will go prow on to use its lances. At 45cm range it'll go broadside (either way lance or regular Murder, same doctrine).
The Carnage will go abeam at once at the 60cm abeam band. And have 10wb's all the time.

Since only few enemy fleets have ranges +45cm the enemy will be closing. A win point for the Carnage! If the enemy gets to 45cm. The Carnage gets more effective (+6wb) and still fire at prows. Yay.

So: Murder NEEDS to close. The Carnage does NOT NEED to close. Carnage will wait to see enemy closing.

A ship that closes is more vulnerable. A ship abeam less.



What would happen in a 1:1 duel?

[left of west]

A point worth noting:

If the lance-variant Murder has broadsides roughly equivalent to the Carnage (I'm not entirely convinced it is, but I'm working up the numbers), the Inferno cruiser in the new Powers of Chaos pdf is almost certainly better than either ship--ten points to swap the lance-murder's forward lances for fp 6 lfr 60cm batteries pretty clearly seems to be a good buy.

The Firestorm's Broadside is identical to the Lance-Murder's, except that its prow battery can participate as well, meaning that its functional broadside is strictly better (by about 50%) than the Lance-Murder's.  Its prow battery stacks up like this:

At all ranges and aspects, the Lance-Murder gets 2 dice at 4+ (expected return: 3/3)
Probability of 0 hits: 25%
Probability of 1 hit:  50%
Probability of 2 hits: 25%

At >30cm to 60cm, closing
Carnage/Inferno get 3 dice at 5+ (expected return: 3/3)
0 : 29.630%
1 : 44.444%
2 : 22.222%
3 : 3.704%

At >30cm to 60cm, abeam
Carnage/Inferno get 1 die at 5+ (expected return: 1/3)
0 : 66.667%
1 : 33.333%

At >15cm to 30cm, closing
Carnage/Inferno get 4 dice at 5+ (expected return: 4/3)
0 : 19.753%
1 : 39.506%
2 : 29.630%
3 : 9.877%
4 : 1.235%

At >15cm to 30cm, abeam
Carnage/Inferno get 2 dice at 5+ (expected return: 2/3)
0 : 44.444%
1 : 44.444%
2 : 11.111%

At <15cm, closing
Carnage/Inferno get 5 dice at 5+ (expected return: 5/3)
0 : 13.169%
1 : 32.922%
2 : 32.922%
3 : 16.461%
4 : 4.115%
5 : 0.412%

At <15cm, abeam
Carnage/Inferno get 3 dice at 5+ (expected return: 3/3)
0 : 29.630%
1 : 44.444%
2 : 22.222%
3 : 3.704%


The Lance-Murder is significantly better against abeam targets outside of 15cm
The Lance-Murder is slightly better against abeam targets within 15cm
The Lance-Murder is slightly better against closing targets beyond 30cm
The Inferno is significantly better against closing targets within 30cm

So, all in all, a moderate advantage to the Lance-Murder in terms of forward-arc shots and a significant advantage to the Inferno in terms of broadside-arc shots.  Really, you guys should probably be suggesting the Inferno to newbies, over either of the older ships.  It's clearly better than the Murder, and I imagine it stands up to the Carnage well, too.  

[Sigoroth]

I see the word 'average' tossed around here all the time with reference to the expected number of hits a particular battery of weapons will cause.  For example, a player gets to roll 6 dice to hit a target with 5+ armor.  Based on this situation, most if not everyone, around here would say that such a shot would average 2 hits.  Eg..  There are 6 dice, and a 1/3 chance of both dice hitting, therefore an average roll would be 6x(1/3)=2.  This average is called a 'mean' average and, although easily and quickly calculated, it's somewhat misleading. First of all, results of dice rolls that have odds of less than 50% (like 1/3) do not follow an even distribution and are skewed towards failure.  Each individual dice has a greater chance of failure than success and that doesn't change if you roll 2 dice or 12.  Averages based off of means can be misleading this way.  It's like if you averaged the amount of money 4 men have when 3 of them only have $1 and the 4th has $100.  You'll get a skewed view of reality.  Mean averages aren't the best way to figure your odds when we are dealing with dice.  Using them to calculate the expected damage output of weapons batteries tends to give the player a feeling that their WBs never live up to expectations.  Second, the mean average doesn't really give you enough info to make a good tactical decision.  For example, what are the chances of missing completely?  What are the chances that I will roll more than 2 hits?  Knowing these probabilities can give you the tools you need to use your fleet effectively.

They're not "skewed toward failure", they are actually right skewed, which, if anything, is skewed towards success. The chances of multiple hits is what drags the mean up after all. I'm sorry, what's the chance of a Murder doing hull damage to an enemy cruiser again?

3 dice 5+ armor
0 hits = 29.63%
1 hits = 44.44%
2 hits = 22.21%
3 hits = 3.7%

4 dice 5+ armor
0 hits = 19.75%
1 hits = 39.5%
2 hits = 29.62%
3 hits = 9.87%
4 hits = 1.23%

5 dice 5+ armor
0 hits = 13.17%
1 hits = 32.92%
2 hits = 32.92%
3 hits = 16.45%
4 hits = 4.11%
5 hits = .41%

6 dice 5+ armor
0 hits = 8.78%
1 hits = 26.34%
2 hits = 32.92%
3 hits = 21.94%
4 hits = 8.22%
5 hits = 1.64%
6 hits = .13%

Yeah yeah, you're not telling us anything we don't already know. We went through this whole thing with the discussion about probabilities of destroying Eldar escorts vs normal ones. Yes, we know that expected average hits isn't the whole story, that the chances of getting a certain number of hits counts too. However, it doesn't really matter in the case of the Carnage/Murder debate. The Carnage is your main gunship. You fire them first. That's why it's clean space. Your other ships often have to deal with BMs, unless you've got a squadron of, say, 3 Devs inside 30cm and a good firing solution on a secondary target. Then they may get a clean shot too.

What you want with your main gunship is the chance of spectacular success. If two systems have the same average number of hits (say, locked on Murder/Carnage against closing 6+ armour at 60cm range) then while one may have a lower chance at, say scoring 1 hit or 2 hits, it will have a greater chance of scoring 3 or more hits. This increased chance of higher damage will actually make up for the reduced chance of lower damage, because the averages are equal. If it didn't make up for it then it would have a lower average. In BFG there are 2 mechanics that actually favour the spiky damage. One is its unpredictability. This makes it harder for the opponent to predict the outcome and therefore harder for them to predict whether BFI is worthwhile. Therefore choosing whether to brace or not approaches blind guess level effectiveness. The other mechanic that favours spike damage is shielding. If you consistently put out 2 damage per turn against a 2 shield target you will never destroy it. If you have a ship that varies 1, 2, 3, 1, 2, 3 then it will eventually destroy the target. One that varies 0, 2, 4, 0, 2, 4 will destroy it quicker and one that varies 0, 0, 6, 0, 0, 6 will do it even quicker again all the while all of these do the same average damage.

That is not to say that reliability of fire isn't useful. It is, when you need to do one more hit to reach a crippled threshold, or you need to destroy that last escort before it can disengage and deny you VPs or the maximum damage the target can soak is limited (say 1 or 2 hits left) then lances really come into their own. Lances also have an advantage when you're out of position, or your only target is behind BMs, etc. Therefore lances are a support weapon. Primary gunships should, all else being equal, make use of WBs. Also, as a side note, pure ships are better than hybrids.

One thing to notice about weapons batteries is that the number of average hits doesn't go up proportionally to the number of dice that you add.

Er, yes it does. For every 1 dice you add, you add 0.33 hits against 5+ to the average. That is a linear proportional increase. Against 6+ armour each dice adds +0.17 hits to the average. Again, linear proportional. Against 4+ armour, you guessed it, another linear proportional increase of 0.5 hits.

I believe that this goes some of the distance in explaining why some players prefer lances to WBs (or Murders with lances to Carnages).  As they always hit on a 4+ they tend to be much more reliable than low numbers of WB dice and also a lot more predictable.

And understanding that the sense of disappointment that people feel is simply a misunderstanding of how probability works, not an actual performance issue with the Carnage or WBs in general why then don't you acknowledge that the Carnage is far and away the better ship and recommend it, with the note that it may seem to under perform but in the long run will actually do better than the Murder?

Also, I have no idea why you keep comparing the Carnage to an abeam ship at long range. The vast majority of fleets will have closing 5+ or 6+ prows to shoot at. There are extraordinarily few circumstances where a Carnage will be forced to shoot at an abeam ship at long range. Also, I don't know how you're going to get the Murder to 45cm and maintain that range bracket. The Carnage can't maintain a 60cm range bracket against other fleets, so I don't see why you expect to be able to achieve it with the Murder.

[horizon]

At >30cm to 60cm, closing
Carnage/Inferno get 3 dice at 5+ (expected return: 3/3)
0 : 29.630%
1 : 44.444%
2 : 22.222%
3 : 3.704%

At >30cm to 60cm, abeam
Carnage/Inferno get 1 die at 5+ (expected return: 1/3)
0 : 66.667%
1 : 33.333%

What the .... fock?

The Carnage vs a closing target above 30cm has 8 dice. Not 3 .
The Carnage vs an abeam target above 30cm has 3 dice. Not 1.

[left of west]

That's a forward arc comparison, not a broadside comparison, Horizon--the point was to compare the forward arc of the Inferno (identical to the forward arc of the Carnage) to the forward arc of the Murder, since the Inferno's broadside is strictly better.  

I stated that quite specifically--gotta read a little more carefully, there, Horizon.  =P

[horizon]

Perhaps I should read better but why on earth should I fire the prow only from a Carnage? I cannot comprehend that...

You need to compare total ship capability. That is that the Carnage prow weaposn swivel left front right and add strenght to the long & medium range broadsides.

[Sigoroth]

@LoW

The Inferno is part of a document that I reject in its entirety. I would not recommend any ships from that document to anyone, as I would not play with or against any of them myself. Also, the profiles aren't identical. The Carnage gets a total of 10WB at 60cm vs the 6 of the Inferno. Due to poor WB scaling with range the Inferno is better than the Carnage in the 30-45cm range bracket, and has parity in the 15-30cm bracket, though is inferior in the 0-15cm and 45-60cm brackets.

That's a forward arc comparison, not a broadside comparison, Horizon--the point was to compare the forward arc of the Inferno (identical to the forward arc of the Carnage) to the forward arc of the Murder, since the Inferno's broadside is strictly better.  

I stated that quite specifically--gotta read a little more carefully, there, Horizon.  =P

It is a pointless comparison however. Pointing your prow at the enemy is asking for your ship to be blown apart for a Chaos fleet and since the Carnage and the Inferno can simply go abeam and still fire their prow weaponry then this is what you do.

[left of west]

@LoW

The Inferno is part of a document that I reject in its entirety. I would not recommend any ships from that document to anyone, as I would not play with or against any of them myself. Also, the profiles aren't identical. The Carnage gets a total of 10WB at 60cm vs the 6 of the Inferno. Due to poor WB scaling with range the Inferno is better than the Carnage in the 30-45cm range bracket, and has parity in the 15-30cm bracket, though is inferior in the 0-15cm and 45-60cm brackets.

I see I guess I thought that was generally considered to be a cannon document.  Oh well.

Also, I never said the Carnage was identical--I said that the broadside-only guns on the Lance Murder are identical to the broadside-only guns on the Inferno, which means that the Inferno (with its lfr prow battery) has strictly better effective broadsides.  I haven't offered a comparison between the Inferno and the Carnage.

It is a pointless comparison however. Pointing your prow at the enemy is asking for your ship to be blown apart for a Chaos fleet and since the Carnage and the Inferno can simply go abeam and still fire their prow weaponry then this is what you do.

It wasn't a comparison between the Carnage and the Inferno--it was a comparison between the Inferno and the Lance-Murder.  The Inferno's effective broadside is strictly superior to the Lance-Murder's (and its broadside-only armament is identical), which means that the comparison between the Lance-Murder's forward armament and the Inferno's forward armament (which happens to be identical to that of the Carnage) certainly does have a point--it's the only point on which the Lance-Murder can hope to offer competition against the Inferno (except price). 

Even if the Carnage never intentionally sets itself up for forward shots, they do have the potential to arise, and they're only one of the three relevant points of comparison between the Lance-Murder and the Inferno whose conclusion isn't obvious. 

This should address Horizon's observation, too. 

Again, you guys need to read what I wrote a little more carefully.  =P

[left of west]

Just in case Horizon still can't figure out what I'm trying to say, here's a response to your post, directly:

Perhaps I should read better but why on earth should I fire the prow only from a Carnage? I cannot comprehend that...

You need to compare total ship capability. That is that the Carnage prow weaposn swivel left front right and add strenght to the long & medium range broadsides.

You're still not reading carefully enough.  I'm not firing the Carnage's prow weapon by itself.  I'm comparing the Inferno's forward armament to the Lance-Murder's forward armament, because it's the only point on which the two differ where one isn't strictly superior to the other--and that makes it critical to the overall comparison of the two ships.

Also, I know that the Carnage's prow guns swivel.  So do the Inferno's.  Obviously =P