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Thursday, April 22, 2010

The madness behind the math...



I'm sure we all like to use mathammer (Dethtron does - I think he made me do this really). It's great for getting a bare understanding of what a unit does and where it's niche lies if you can't look at it and figure it out. I've crunched the numbers myself many a times to dazzle with amazingly small (or large) fractions but quite frankly, it's lacking. It's basic statistics and is a probability which doesn't take into the account that in the real world, we are can only tend to a probability of 1 (this means nothing is ever certain, sorry Newton!). Quick example. 6 Fire Dragons unload all their meltaguns onto an AV10 tank. 4 hit. All 4 are highly likely to pen. You should kill the tank twice. How often has this NOT happened to people? Stastically with an AP1 gun you want 2 hits, with non-AP1 you want 3 hits. How often does it take 10 hits? The raw numbers are great to give you an idea about a unit's capabilities but unless you understand statistics or look at where the numbers are coming from, spouting this sort of mathammer makes you look like an idiot.
So why on earth am I telling you this? I'm sure none of you want to hear about math nor anything about inferential statistics (psychologists as a general rule of thumb hate statistics because they are people persons. I love bucking the trend). Here is a direct link to a post I made about the numbers on a TL-AC compared to an LC or TL-LC against high AV and one to the original post.

So whilst that covers more about the math, etc. Let's look also at opportunity cost. The TL-AC is a much more reliable platform against AV10 than the Las/Plas. I don't care about the numbers here because I know how 40k and dice work. The TL-AC has 4 re-rollable S6 dice. The Las/Plas has 2 re-rollable S7 and 1 S9. The TL-AC is more reliable but the Las/Plas has more utility further up the AV chart and is not 'bad' against lower AVs. After all, how many armies survive on S6/7? Oh right, a lot. However, shooting the Las/Plas at lower AV loses you out against medium AV. Just liking shooting a meltagun at lower AV loses you out against higher AV. This is something I think most people in that post are missing. Who gives a flying hoot if my TL-AC or Las/Plas RBack is better against AV14? They both suck at it in turn when you compare what you are losing. Now, if you have nothing else to shoot or taking out the AV14 wins you the game (i.e. it's contesting or something), go ahead. Shoot it and cross your fingers because whilst the raw stats give you a 5% or 3.7% chance of destroying it, you've got so many dice to go through it's not funny.

This is what people don't understand also. The more times you have to roll, the less likely something good will happen. Look at my Fire Warrior and Marine example in the link. According to someone like RCgothic, they are the same. According to statistics they are not because the Marines are more likely to land significantly more hits whilst the FW are more likely to land significantly less hits which then affects the wound rolls. Over 200 billion games they may have caused the same amount of wounds but if you have one chance to take out a squad I'm going to prefer the higher to hit roll first as it skews the statistics in my advantage. This is why twin-linking is better than ignoring cover, why rending is nerfed, etc. The more chances you have to affect dice rolls further down the chain, the better. Hence rate of fire being so important in 5th edition. If I land 5 hits I'm likely to cause SOME damage. It may not kill but it's at the minimum going to stop it shooting and/or moving.

A final example pertaining to opportunity cost and the numbers. The numbers say that my Railgun is just a tiny (like 1/50th) more effective than a meltagun at close range on AV10. However, the meltagun is the better gun because that railgun is a much, much better anti-infantry gun with its submunition pie plate. In the case of the TL-AC and Las/Plas RBack shooting at AV14 is losing you out on some pretty reliable low AV shooting or anti-infantry fire.

9 pinkments:

GWvsJohn said...

I hate to do this to you Master Kirby, but your Marine/Fire Warrior example is just plain wrong. 10 Marines Rapid Firing Bolters at t4 (3+ then 4+) vs 10 Fire Warriors Rapid Firing Pulse Rifles (4+ then 3+) has the exact same distribution curve. Unless something like rending affects how the system works, the order really has no effect. I'm sure all the math faculty at Penn would cry that I can't explain it myself, but if you don't believe me, PM Volandum on Warseer, it's like his pet peeve.

MasterSlowPoke said...

"if you have one chance to take out a squad I'm going to prefer the higher to hit roll first as it skews the statistics in my advantage. This is why twin-linking is better than ignoring cover, why rending is nerfed, etc."

It doesn't. Two dice rolls are independent events, and independent events cannot affect each other. It might feel more likely but the result is the same.

Think of it as a single die. If you hit on a 3+ and wound on a 4+, it is totally irrelevant what order you roll the hit and the wound. For a single die, the concept is pretty ridiculous, right? When you're rolling 10 dice it seems like rolling the higher chances first is better, but rolling 10 dice at once is no different than rolling a single die ten times (we will ignore measured rolling here).

I play Guard, and I'm constantly firing BS3 weapons at vehicles with 4+ cover saves. Which order do I choose? Bring it Down or Fire on my Target? It doesn't matter - they both do the same exact thing, statistically.

Rending was nerfed because old rending removed the to-wound roll entirely. That actually does make later rolls better.

Kirby said...

The point here is not the number at the end, it's the affect of the numbers before on the end. Rather, it's how we get to that number. If I can get a 50% chance of doing something off 10 dice, it's not as good as a 50% chance off 1 die (which is the point of the original post on Warseer).

I know Bring it Down and Fire on my Target at the same time as SM/FW have the same end result, X wounds or X% to kill tank but you want to look at the spread of scores and end result Odds Ratios rather than the statistical number.

Without explaining a bunch of math and recalling the TFex example. There are 3 possibilities there and only one possibility has you not hitting. This spread with BS4 and rapid firing boltguns is a lot closer to one, same with the FW but the Marines are closer to 1 than the FW. This makes them more reliable. Whilst this is flipped around in the wound stage because the Marines are more likely to have more hits they are more likely to skew the wound results in their favor.

Again, over hundreds and billions of games, FW and SM will perform the same against T4 targets. Just like if you drop a ball it will go straight down. Sooner or later though, it's going to go up because of some random event (i.e. my foot).

The whole point of this isn't to say SM > FW but rather the more reliable you are at the beginning of your rolls and the less dice you have to roll to get to your end result, the better (i.e. no rending).

And rending was nerfed because it was on to hit, not that big an effect on Assault Cannons but on Quins who have 40 attacks and 20 hit? They had 40 chances of a 6, now they have 20.

Anyways does that make more sense now? Call it quantum statistics or random chance, etc. but the standard deviation spread does impart a lot on dice rolls. Look at crisis suits. PR/MP have a huge spread early on over TL-MP but on average they hit the same. However, PR/MP have an infinitely better chance of getting more hits than TL-MP can (add can be bolstered by ML; same with FW).

Making your earlier rolls impact more heavily on your later rolls is more likely to create statistical deviation and push you up into the 90th+ percentile which considering average rolls for the rest of your shots leaves you with a bigger impact thatn before. I'm gonna go to bed now and see what results from this.

GWvsJohn said...

That's not true Kirby. Not only do 10 SM and 10 FW have the same mean, they also have the same standard distribution and same chance to get any particular # of hits. In my heart of hearts I agree that it seems like the 3+ on the hit roll makes a higher # more likely, but it doesn't.

MasterSlowPoke said...

A 3+ followed by a 4+ is exactly the same as a 4+ followed by a 3+. It does not matter if you roll 1 die, 10 dice, 2 billion dice. The results will be exactly the same in every case - this is the communitve property of multiplication. No "quantum" mechanics apply here. You're not affecting the 4+ rolls by rolling the 3+ first.

This is how rending used to work, say against MEQ:

1/6+3/6*2/6*2/6 = 2/9 chance to wound per hit

Now it is:

4/6(1/6+1/6*1/3) = 1.332/9 chance to wound per hit

It's more that they can't avoid the to-wound funnel than that they have less chances of rolling a rend. If, for example, rending was still on the to-hit roll but you now had to roll to wound (essentially making 6's to hit power weapons), the result would be identical to the current rending system, even though you have have far more chances to roll that rend.

MasterSlowPoke said...

Actually, I got that last paragraph wrong. If you still rended on the to-hit roll but still had to roll against toughness, you'd actually be worse off against MEQ. Even though you have more chances roll a 6 and ignore that armor, you're only sticking a rend 1 in 18 attacks, where in the current system, you stick a rend 1 in 9 attacks. In 4th edition, rends stuck 1 in 6 attacks. I probably just confused everyone so my bad, ignore that part.

Kirby said...

On the drive this morning I figured out the discrepency here :P. I'm basing everything of inferential statistics which is based off of samples, not population parameters. Here we actually have the population parameters (something inferential stats generally doesn't have because it's impossible to ask everyone). Marines hit and wound on X and so do FWs.

Again, over the X number of games they are going to be the same but if you take any given game one of them is likely to perform better or worse than the other (back to the TFex example, not every game is going to have 6 hits from the TFex over 6 games).

Here is where our discrepency comes in. Let's assume hitting on a 4+ equals a normal distribtuion. Therefore hitting on a 3+ is going to be skewed left (the bulk of the results are in the right hand side of the curve). Now SM and FW use both of these curves so over X amount of time there will be no discernable difference between them. So our null hypothesis here is that SM = FW, which is true and thus we accept it.

So because our original SM population is skewed, we are more likely to make an error (all of inferential statistics is based off of assuming a normal population) of the type 1 type (so we reject the H null when we shouldn't). Whilst this is also likely to happen for the FWs on the to wound roll we have front loaded the SM statistics.

Again, I agree with you that over a period of time they are no different and really what I'm saying is a bunch of theory (I wrote on paper on what I called front loading. So many psych papers have barely significant results and screw up their statistics by not taking random samples, priming, etc.).

Digressing, sorry ^^. I apologise for the confusion, I'll take my stats theory and inferential statistics and hide it where no one can see it, yes? Though my original point still stands about the AC/LC debate though =D.

Oh and SlowPoke; I don't have my old 4th ed book on me but correct me if I'm wrong. Old rending was any rolls of 6 auto-wounded and with the AP2 result. Today it's any wounds of 6 cause the 'rend.' Since you are rolling more dice on the to hit end of the scale you are more likely to get a 6 from that.

GWvsJohn said...

Yikes, I didn't understand any of that :)

Are you saying you agree that there's no difference? Or that there is, but it doesn't matter?

Unless something affects or depends on one of the rolls (like rending) a 4+ then 3+ is the same as a 3+ then 4+

Kirby said...

I always agreed there was no difference :P but with complex theory mongering later blablablablablabla lol.

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