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Lowinor wrote:A 3+ 4+ sequence is exactly identical to 4+ 3+ in expected, modal, and median results. It doesn't actually make any difference which order the tests are taken in since all tests have to succeed.

Exactly. Nurglitch has created his own branch of statistics.

He was trying to say a 3+ then 4+ is different to a 4+ then 3+. That is actually wrong.

tzeentchling wrote:Again, though, the probability and averages do cancel out... if you have a large enough sample. As Rosicrucian's plots show, when the sample size is large, things will average out to the most probable numbers. 40K does not have that large sample. Here, deviations from the norm dominate over the norm. I imagine if Rosicrucian redid his plots for sample sizes of 10,50, and 100, the comparison would not be as close, and the plots would probably differ from run to run.

This is true, however there is equal weighting of one being better than the other.

The plots will differ but the fact remains the same, it doesn't matter which one you pick it's the same probability of killing marines. They are equal in distribution.

You could do it for 50 runs, get a plot that makes cover saves look better, then do another 50 runs and BS looks better. Either way it's equal chance of either looking better.

Lowinor wrote:1) 6 pulse rifle shots, first against ork boyz in the open, second against terminators.

(1/2*6d) + (2/3*(1/2*6d)) + (1*(2/3*(1/2*6d))) = 7d

(1/2*6d) + (2/3*(1/2*6d)) + (1/6*(2/3*(1/2*6d))) = 5.33d

You're telling me that the difference between ork boyz and terminators, in your "utility" calculation is 7 to 5.3? In reality, you kill six times as many boyz.

This post is made of win.

Lowinor wrote:For extra credit figure out what happens in your formula if you put in an automatic fail anywhere except the first term; you end up with non-zero "utility" for a sequence of rolls that will never provide results. The reason for this is you're calculating total successful rolls instead of something actually useful.

Using nurglitch's calculations, the number I get when I shoot 5 bolters at a wraithlord is 3.33.

So I get 3.33 with 5 bolters against a wraithlord. Fantastic. However knowing that is completely useless, because I do 0 wounds to the wraithlord every single time.

onlainari wrote:I never actually used average in any of my posts.

Average is a rather poor man's mathematical tool.

In the context that you mentioned- predicting the outcome of a given situation- average is a poor tool.
However, when it comes to comparing the effectiveness of two different situations (like the question posited by the original post), it's perfect.

Not making a dig at anything you've said, just pointing it out.

Here's a tricky question.

I want to shoot a solid shot railgun at a marked space marine in cover.

I can either increase the ballistic skill or reduce his cover.

Theoretically, it's either

(5/6)*(5/6)*(1/2) = 0.35

or

(2/3)*(5/6)*(2/3) = 0.37

chance he dies.

Sounds like you would use it to reduce cover right?

However I have to use the markerlight before I roll to hit.

If I use the markerlight before I roll to hit to increase BS, then I'm getting use out of it, and have a 0.35 chance to kill the marine (rather low for a railgun eh? oh well).

If I use the markerlight before I roll to hit to decrease cover, I may not get a use out of it. So, does that reduce the probability of killing the marine?

No. It's still 0.37.

This is rather odd and difficult to think about. I have reduced a marine's cover, but I failed to hit. So reducing his cover did nothing. So how is it still 0.37 to kill the marine?

What if I could wait until after I wounded? What's the chance of killing the marine then? It's still 0.37 of course. However, I only have a (2/3)*(5/6) = 0.56 chance of using the markerlight.

So in effect, I am losing 0.44 markerlights by reducing the cover save. This only matters if I had other things that can use markerlights.

It's something that's far too complicated for me to think about. But basically, using the markerlight to reduce cover does not have an effect on the chance the marine in killed. In other words, using the markerlight to reduce cover of the orks, instead of increasing the ballistic skill, when I'm shooting 12 fire warriors at them has no effect on the number of orks I kill. It has an effect on the cost in markerlights.

Nurglitch wrote:I'm not arguing the statistics.

Then you're babbling -- the results of 40k dice rolls are very simple to model mathematically and anything that's better than something else can be quantified in some manner or another.

You have yet to quantify anything, other than giving a formula for calculating expected successful rolls and asserting (incorrectly) that it means something.

I'm simply making the point that given the algebraic structure of the shooting operation in 40k that there's more to it than just the statistics.

If there's something there, it would be quantifiable.

If we ignore the fact that saving depends on hitting and wounding, and we ignore the fact that wounding depends on hitting, then sure, there's no difference between the cumulative result if we only consider the likelihood of the cumulative result obtaining.

And you ignore the fact that the utility of hitting depends on the chance of wounding and failing saves. The reliance of the separate events goes both ways.

Your assertion that "There's no difference in effect between "roll a die, and if it's 3 or higher, roll another die, and record a wound if it's 4 or higher" and "roll a red die and a blue die, and if the red die is 3 or higher and the blue die is 4 or higher record a wound"." is true if and only if we're talking about the likelihood of a wound obtaining.

It is false if we are talking about maximizing the number of wounds obtaining by maximizing either the expected value of hits or the expected value of failed saves.

Ok, so, what you're saying is, in a nutshell, there's a difference between the following:

1) Roll a die; if it is 3 or higher, roll another die. Count it as a success if the second die is 4 or higher.

2) Roll a red die and a blue die. If the red die is 3 or higher and the blue die is 4 or higher, count it as a success.

Just so we're clear -- you're arguing that the two do not have absolutely identical results?

Still, I gotta admit, I do like the idea of rolling all my hit, wound, and save dice at once and cherry picking the results!

You need a hell of a lot of differently colored dice to pull it off, but feel free -- the results are exactly the same. The reason for rolling dice in batches is to smooth out game play; if you roll hit/wound/save simultaneously, you need each individual die to be identifiable; if you roll all hits, then all wounds, then all saves, you don't. You end up rolling less dice too, but the results are exactly the same.


In any case, you repeatedly claim things that, if true, would be quantifiable. You haven't quantified anything -- you've given a formula which you claim provides "utility", but have provided no way to translate that "utility" to reality. If there's an advantage to 3+ 4+ over 4+ 3+, it's measurable. Measure it for us.

onlainari wrote:If I use the markerlight before I roll to hit to decrease cover, I may not get a use out of it. So, does that reduce the probability of killing the marine?

One of us is missing something here

In either case, you're getting something out of it. In the three cases relevant (Railgun at marine, BS+1 Railgun at marine, Railgun at cover-1 marine) each has a different success table -- the first is successful on (3+, 2+, 3-), one on (2+, 2+, 3-), and the other on (3+, 2+, 4-).

Effectively, we're rolling three dice and looking up the results in a table -- 216 unique results, and each has a different set of successes on the table. The default setting has 60 (4 * 5 * 3) successful slots, the second 75 (5 * 5 * 3), and the third 80 (4 * 5 * 4). The 15 extra slots you get from the BS boost are different than the 20 you get from the cover reduction, but all have an equal chance of showing up.

Another way of looking at it is with decreasing the cover save, you aren't increasing the chance of a hit, but instead increasing the value of a hit. You still benefit, as you're getting a chance at a better hit instead of a better chance at a hit.

olaniari:

You're only reducing his probability of having to roll the die. That isn't changing whether or not the markerlight's value, since in the situations it IS used you have a large advantage.

To think about all of this it's important to understand that in any case where there isn't something special going on (like rending), it does not matter what order you roll the dice.

If you wanted to do everything where you first rolled to wound with ALL the dice, then you rolled to see if those wounds actually were hits, then rolled to see if those hits were saved, that would be fine. You get the same result every time. This is what happens when you have probabilities that are multiplied

You're just looking at the same problem as before with the fire warriors, but for a single firer.

Nurglitch wants to believe that whatever you do first matters most. The idea being that you can't wound if you don't hit, and you can't force a save if you can't wound. Unfortunately this doesn't actually happen, and all those extra dice get annihilated by the power of probability in the end. They end up having to roll more dice, but well it didn't matter, since they (probably) made more saves. But it definitely looks scarier for them. And, if you don't understand probability, and believe that your previous results matter and weren't just a calculation that we've now moved on for since we have an answer, you think you've somehow improved your odds by giving yourself better chance of success early, in tradeoff for a better chance of failure later.

Anyway, I'm probably not making sense. The key is again that the order does not matter, you can always do all of these operations any which way you want, so long as the probabilities stay the same the results stay the same. Feel free to always roll their save first - it will make it feel like the markerlight "did" something, and have no effect on the final result, statistically.

lambadomy:

I think you've hit the nail on the head as to how my point is getting completely missed here. You say that I want to believe whatever you do first matters most. That's half-right, I don't particularly want it that way, but that's the way it crumbles when you're dealing with embedded conditionals where order matters rather than a series of conjoined operations where the product of the probabilities is the same in any order.

You also say that I think I've somehow improved my odds by giving myself a better chance of overall success by improving the odds of the earliest operation. That should be plainly false.

I don't believe that, as I've agreed that the calculation of the probability is associative. What I do think is that one can improve the results, whatever they are, by giving oneself better odds at the earliest point in the operation.

That is because the weighting determined by the order is not the probability of any particular result turning up on the dice, but by the number of possible results, the number of dice you roll.

A greater number of dice has greater utility, and since the to hit/to wound/save process is a sorting operation, you will always roll more than or equal to the same number of dice in earlier operations.

That's why the probability is beside the point: we already know that the total probability of any wound happening is the same whatever the order the dice are rolled. But since we can't use that information to magically encourage the dice to roll the way we want them to, and it's the same either way, it's irrelevant. Modifiers don't change the probability of result showing up, as it would be the same overall result regardless of which sub-operation you apply the modifier to.

What's relevant is what we can change: the expected value of either rolling to hit, or the expected value of rolling to wound. And since the probability part of the expected value is irrelevant, what's important is the expected utility of each operation.

The utility is the number of dice, and thus the greatest potential number of wounds (ignoring, of course, Instant Death, etc). You want the modifier to apply to the most dice possible, which will either be the hit dice, because each to hit die is a possible wound, or simply be irrelevant since modifiers change the results of the dice themselves - hitting on a 5+ and adding +1 to your results is irrelevant when you roll a natural 6.

The fact is that the order matters because there is more to Warhammer 40k than mere statistics. I think that's where a lot of players fail as effective players because they think that they can judge shooting or close combat depending on the more likely outcome, rather than weighing all the possible outcomes by their probability of occurring.

That's why I reject the proposition that Possessed Chaos Space Marines are random: You know that you'll get 30 possible combinations of special rules, if you buy an Icon , and since they'll all equally likely, you might as well plan for the eventualities rather than whether they might happen - you already know they might happen. Just like we already know that if you roll one die at 3+ and one die at 4+ the odds of getting both results will be exactly the same as rolling one die at 4+ and then one die at 3+.

That's why you should roll the dice in order, as specified in the rules, rather than dicking around the order, because the number of dice you roll depends on the order! And the most dice you will roll, if the markerlight is ever relevant (i.e. when the dice results aren't naturally successful), will be the to hit dice!

Take, for example, rolling ten to hit dice at 3+ to hit, 4+ to wound, and 2+ to save. If you just check the odds of something happening, you can roll them in any order. Except when a weapon ignores armour, or its AP < Sv, or when rending is an issue, or you know, basically whenever you're playing Warhammer!

I mean, of course the order you are rolling them in is statistically irrelevant. So what? That was pointed out at the beginning of this thread, and no one disagrees with it. I certainly don't. What I disagree with is the relevance of statistics when it has been pointed out that the overall stats don't. That's what's sometimes called a 'ceteris paribus' principle, assuming that if something is invariable, that you can safely ignore it.

Given that we can safely ignore the statistics of the matter, that leaves us with the formal aspects of classical game theory, namely utility, which is variable over the course of the entire operation, and given that the topic is about which sub-operation is weighted such that we prefer to modify its probable outcome, then it is entirely relevant and in fact correct to point out that order matters.

Nurglitch wrote:Take, for example, rolling ten to hit dice at 3+ to hit, 4+ to wound, and 2+ to save. If you just check the odds of something happening, you can roll them in any order. Except when a weapon ignores armor, or it's AP2

Incorrect. You can roll the hit, wound, and save dice in any order, even if the weapon ignores save.

Otherwise it appears you're no longer saying "increase BS because you kill more". I'm not sure if you were ever saying that but that's what it sounded like.

Generally, when you shoot, you only get utility when you kill models. You say you get utility out of rolling dice, well I'm just not that much into rolling dice.

For simplicity, lets say there are only two rolls here:

Your to hit, their to save if you hit.

You can choose between either a .01% chance to hit and a 99.99% chance to wound, or a 99.99% chance to hit and a .01% chance to wound.

If Nurglitch is right, then this case should be one where we see an obvious difference and you should definitely choose to 99.99% chance to hit!

But again, it ends up being pretty obvious it doesn't matter.

Anyway, this has been beaten to death. you can roll the dice in any order, or all at the same time, and determine successes along the chain, and it will not matter.

The Tau ability to reduce cover saves instead of getting +1 BS is probably just so you can do really expensive Seeker missle assassination sniper shots into cover and so that Shas'o doesn't feel left out.

That, and the only time it would definitely make a difference is for twin linked weapons.

I think the only reason why you couldn't completely reverse the order of the dice (roll saves, roll to wound, roll to hit) is because of wound allocation in complex units. How about that for introducing realism and drama into the game? 5 man squad gets shot 5 times, anybody could die...

Lowinor wrote:Sigh.

So, what's the unit of your formulae? What do they actually calculate? You're multiplying numbers and then adding them when adding them is meaningless. You still don't seem to understand what your calculations actually yield. The only thing they actually do calculate is successful die rolls, which as I've said isn't particularly useful.

Just to explain that your calculations are pointless, allow me to give you two examples:

1) 6 pulse rifle shots, first against ork boyz in the open, second against terminators.

(1/2*6d) + (2/3*(1/2*6d)) + (1*(2/3*(1/2*6d))) = 7d

(1/2*6d) + (2/3*(1/2*6d)) + (1/6*(2/3*(1/2*6d))) = 5.33d

You're telling me that the difference between ork boyz and terminators, in your "utility" calculation is 7 to 5.3? In reality, you kill six times as many boyz.

2) A more abstract example; a sequence of 1+ 4+ compared to 4+ 1+, using "6d" again:

1+ 4+: (1*6d) + (1/2*(1*6d)) = 9d

4+ 1+: (1/2*6d) + (1*(1/2*6d)) = 6d

Please note -- in the above situation, you roll the exact same dice in both cases, but by your math the utility is different.


For extra credit figure out what happens in your formula if you put in an automatic fail anywhere except the first term; you end up with non-zero "utility" for a sequence of rolls that will never provide results. The reason for this is you're calculating total successful rolls instead of something actually useful.

Slight necro but I wanted to comment on this outstanding post. Nicely done.

I think this thread pointed out how wrong 'gut feel' can be.

That's very well said.

Garbage in, Garbage out, no matter how fancy you make it look. A useful life lesson as well.

Okay, let's look at some differences in improving to hit rather than increasing saving throws.

Suppose there's normally 24 attack dice, hitting on 4+, wounding on 4+ and cover saving on 4+. That means, on average rolls at each step of the way we'll get 12 hits, 6 wounds, and 3 failed saves.

If you improve the to hit to 3+, then you get 16 hits, 8 wounds, and 4 failed saves.

If you improve the cover save to 5+, then you get 12 hits, 6 wounds, and 4 failed saves.

So, clearly, it makes no difference whether you improve the likelihood of hitting or lower the likelihood of saving. Right?

Wrong.

The thing is that we're not really concerned with the likelihood of one result happening or another, because they're equal. What we should be concerned with is what happens when dice results don't match the averages.

That's the thing about the Warhammer hit/wound/save progression: at each step you usually roll less dice that previously, so the order matters.

If you improve the to hit, you're more likely to have more dice to wound and to force saves. If you improve the saving throw, you're more likely to have fewer dice to wound and to force saves.

The saves will be more likely to fail, but who cares if you already missed?

Well this is kinda scary....such maths is good to see! You all paid attention in class, yes you did!

Also I agree with onlainari and Lowinor on this. Though perhaps onlainari's tricky question about the railgun is where the argument lies....

I think the problem here is a failure to communicate, on my part probably more than anyone else. The thing is that the difference between adjusting the likelihood of hitting and adjusting the likelihood of saving is statistically irrelevant, but that difference isn't what's at issue (or should be at issue).

If, during shooting in Warhammer, you rolled the attack dice, to wound dice, and saving throw dice all in parallel, then it wouldn't matter whether the hit dice were more like to hit, or the saving throw dice were more likely to fail. That's because, like the process of throwing dice that match the ideal statistical distribution, each throw is independent.

However, each step in the hit/wound/save process depends on the previous step, and the following is true if each step is figuring in the number of dice rather than the likelihood any any particular result occurring:

hit >= wound >= save

Suppose 6 dice to hit. That means the maximum dice to wound and save will be =<6.

That means increasing the number of hits increases the potential number of wounds and saves despite not increasing the likely number of unsaved wounds.

Increasing the potential number of wounds and saving throws is good because the die result deviate from the what is statistically likely. Given that die results are not cumulative, any dice you roll are as likely to come up 6s as to come up any other mixture of results.

So you want to open up as much potential as possible, as well as increase the likelihood of the end result.

Let's take the example I gave previous:

12 @ 4+, 6 @ 4+, 3 @ 4+ ├ 3 unsaved wounds

The likely result is 3 unsaved wounds.

Whether we modify the hit step to 3+ or the save step to 5+, the likely result is

12 @ 3+, 8 @ 4+, 4 @ 4+ ├ 2 unsaved wounds
12 @ 4+, 6 @ 4+, 3 @ 5+ ├ 2 unsaved wounds

So the likelihood doesn't change (2 unsaved wounds equals 2 unsaved wounds...).

But notice the number of wound rolls and saving throws.

8 ≠ 6, 3 ≠ 4.

Well, so what right? The end result is likely to be the same. The fact that it's likely to be the same doesn't mean it will be the same and a good Warhammer player isn't just prepared for what's likely to happen, they're also prepared for what can happen.

If the dice all roll 4+ to wound, then there's either 8 or 6 saving throws. If all the saving throws are passed, then they're both equally 0, but if they're all failed, you get 2 more wounds for having improved the likelihood of the to hit roll.

By adjust the to hit roll using Markerlights, you're not only increasing the likelihood of causing some number of unsaved wounds, you're increasing the potential number of unsaved wounds, and more so than if you increased the likelihood of successful to wound an saving throw rolls.

As any Ork player knows: rolling more dice doesn't make you lucky, it just means you get more results when you are.

Take the debate, or lack thereof, over the utility of Star Cannons vs Scatter Lasers. Such, the Star Cannon denies all armour saving throws, but the Scatter Laser can cause twice the potential amount of damage. It's that potential that means people take Scatter Lasers and leave the Star Cannons at home.

So, just to reiterate for the tl;dr crowd: I'm not in any way arguing that buffing the to hit instead of de-buffing the saving throw will make unsaved wounds more likely.

I'm saying that buffing the to hit before de-buffing the saving throw will yield more unsaved wound when you get lucky.