Dice minus seven isn't zero damage. It's one.

[ringkichard]

I used to have a link to this message in my sig, but that link broke with the demise of the old forums. So it seems like it's time to revise my public service announcement for the new generation:

One of the first things gamers learn about dice is that the average roll on 2d6 is 7.* It's true and it's useful to know. But then, one day, your ARM 19 warjack is subject to some POW 12 shooting, and you think, "That's dice-7 for damage. 7-7=0. My Jack is safe." However:

2d6-7 is not 0.

We've all done it. But when we do that, we are wrong, wrong, wrong. That Jack is going to take a moderate beating.

I used to have a long and complicated explanation for this (and I still do; keep reading). But the simplest explanation is that 2d6-7 isn't zero because 2d6-12 is zero.

2d6-7 is actually approximately 1. Which means that your Mangler getting shot at by those Hand Cannons is actually going to take an average of one damage per attack, and probably going to lose a big chunk of its health.

Here, I brought you a graph:


So what's going on here?
In Warmachine, there's no such thing as negative damage. If you roll a 6, and subtract 7 because of ARM, the rules treat that result as 0 damage, not +1 health to the jack. This is why Dice-12 is 0, and not -5.

If your Jack suffers 36 attacks, it will take (on average):
5 attacks for 1 points of damage (rolled a 8)
4 attacks for 2 points of damage (rolled a 9)
3 attacks for 3 points of damage (rolled a 10)
2 attacks for 4 points of damage (rolled a 11)
1 attacks for 5 points of damage (rolled a 12)

Add all that together and you get:
(5x1)+(4x2)+(3x3)+(2x4)+(1x5)=35
And 35 damage over the course of 36 attacks works out to just about one damage per attack roll.

Ok, fine, but what if the example were an ARM 20 Juggernaut instead of an ARM 19 Mangler? Or an ARM 18 Nomad? That graph up there shows a nice curve, but it's kinda hard to read. What are the average damages for ARM-POW?

Here are the actual values:
-00 7
-01 6
-02 5
-03 4.03
-04 3.11
-05 2.28
-06 1.56
-07 0.97
-08 0.55
-09 0.27
-10 0.11
-11 0.03
-12 0

You'll notice that the actual averages start diverging from what you might expect at "dice - 3". That's the first instance of the "zero damage floor" coming into play. If you roll snake-eyes, the rules turn that 'negative one' result into a zero. This continues as you go further down the chart.

The chart is actually fairly easy to memorize, if you round things a little bit.

-00 7
-01 6
-02 5
-03 4
-04 3
-05 2.25
-06 1.5
-07 1
-08 0.5
-09 0.25
-10 0.1
-11 0.05
-12 0

"Bonus Damage" kicks in at Dice Minus Five, and proceeds from there. Even if you just remember that Dice Minus Seven is 1, you're in much better shape than you would be otherwise.

So the next time your opponent complains that "On average rolls, you shouldn't have done any damage, you lucksack!" just smile at him and know that you're winning through the power of math. Or hit him with a high-velocity tubesock-encased pewter bludgeon... whatever. You're justified.

-- EDIT --
"But wait," you cry!
"Dice - 7 does ZERO damage 58% of the time. The average is clearly not 1."

To answer this, we have to talk about the difference between two different types of averages: Mode and Mean.
The mode is the most common value in a group. And it's true, 2d6-7 damage has a mode of 0.
When you roll 2d6-7, you are more likely to roll a 0 than any other number.

What I'm talking about here is a type of average called the Mean, also known as Expected Value.
The mean damage for 2d6-7 is 1 (actually 35/36).
When you're figuring damage to Jacks, Beasts, Warcasters, and Warlocks, you can expect an average of 1 damage per 2d6-7 attack.
Half the time it will be higher, half the time it'll be lower, but on average, that'll be the damage dealt over the long term.

"WAIT! How much higher, and how much lower?!"
Good question, and the answer to that is another whole article.

-------------------
* Yes, it really is 7. Don't be superstitious; it's bad luck.

[quindraco]

You are my favorite forum poster.

I have modified my sig to link to the base index page of the wiki, which links on to the errata and the new page I just made copying Ringkichard's post.

While everyone is welcome to post their own studies of Warmahordes math, Ringkichard is the standard by which you will be measured. You Have Been Warned.

[Septimus]

Oh no! Math!

[solkan]

I declare the original poster a pessimist, and misleading.

The truth of the matter is that Dice Minus Seven is essentially a coin toss, and the average of an event that will go one way half the time and go a radically different way the other half of the time becomes essentially meaningless.

[Nion]

I think the confusion here is that the average damage is 1, but the average attack is zero. In your example

If your Jack suffers 36 attacks, it will take (on average):
5 attacks for 1 points of damage (rolled a 8)
4 attacks for 2 points of damage (rolled a 9)
3 attacks for 3 points of damage (rolled a 10)
2 attacks for 4 points of damage (rolled a 11)
1 attacks for 5 points of damage (rolled a 12)

only 15 out of 36 attacks did any damage. 58.3% of the attacks dealt no damage.

Probability works better when we are sure everyone is talking about the same thing.

[SteakAndSpirits]

Great post.

-s&s

[Zombied00d]

I will have none of your math based chicanary sir! Statistics is a liar and a tool used by liars!

[PG_possiblyarowbot]

THEORETICALLY the average on 2d6 is 7.


My real-world testing shows it to be 5 or less.

[themocaw]

I actually made out a full chart on expected damage values at dice minus X for 2, 3, and 4 dice, as an excel 2007 file. I might post it up if I can find a decent site to share the file.

[EpicAstinos]

THEORETICALLY the average on 2d6 is 7.


My real-world testing shows it to be 5 or less.

Bah! You just haven't rolled enough for your experience to have more closely aligned with what the math predicts it to be!

[quindraco]

I actually made out a full chart on expected damage values at dice minus X for 2, 3, and 4 dice, as an excel 2007 file. I might post it up if I can find a decent site to share the file.

While dice minus X for 2 dice is now done, you are absolutely welcome to put up your data for 3 and 4 on the wiki, or give it to me to do.

[PG_moz]

Your numbers work well for average results when throwing a lot of attacks, but average results are not what you should come to expect from your own models or your opponents attacks! A more effective way to think about the fate of ye old Mangler vs. say 15 Winterguard single-shotting is to sum the damage by probability (important not to incorporate less than zero results!)

http://anydice.com/program/922
15_P12_A19.jpg

The results show your numbers are good for guessing what will happen %50 of the time, but it's critical to think about what happens the rest of the time. Betting on averages is asking to be disappointed half of the time, like trying to win a game of warmachine on flipping coins. I personally like to make the above graphs for common rolls and focus on the expected results compared to the throw of a D6. From the graph:
15 POW 12s vs ARM 19
2+ (83%) : 10 dmg
3+ (66%) : 12 dmg
4+ (50%) : 15 dmg
5+ (33%) : 17 dmg
6+ (16%) : 20 dmg

Going on these numbers, if it's a really important situation I'd only count on 2+ or better results, up to 3-4+ if it's not that important. Leave as little to chance as possible! In anticipating my opponents capabilities I'll usually hedge my expectations of their numbers in the 5+ range. Better to be surprised at how poorly your opponent does than the alternative.

[ringkichard]

I declare the original poster a pessimist, and misleading.

The truth of the matter is that Dice Minus Seven is essentially a coin toss, and the average of an event that will go one way half the time and go a radically different way the other half of the time becomes essentially meaningless.

That would be true if 2d6 was distributed bimodally, but it's not.

The relevant line on the above graph is the blue triangle furthest to the left.

Now we need to talk about the different types of averages: mean, median, and mode.

The mean of 2d6 is 7. This means that when you add all the possible results together and divide by the number of possibilities you get 7.
Also, the median of 2d6 is 7. This means that when you arrange all the possible results in numerical order, the value in the middle of the list is 7.
Also, the mode of 2d6 is 7. This means that 7 is the most common number rolled. There are 6 ways to roll a 7 on 2d6, more than any other option.

When we say, "The average of 2d6 is 7" we can potentially mean any of these things. And they're all true, so that doesn't get us in trouble too often. But this is one of those tricky cases where we need to be able to tell the difference.

2d6 is essentially shaped like a triangle.
We can show this by listing all the possible results sorted by their sum:

02: 1,1
03: 1,2;2,1
04: 1,3;3,1;2,2
05: 1,4;4,1;2,3;3,4
06: 1,5;5,1;2,4;4,2;3,3
07: 1,6;6,1;2,5;5,2;3,4;4,3
08: 2,6;6,2;3,5;5,3;4,4
09: 3,6;6,3;4,5;5,4
10: 4,6;6,4;5,5
11: 5,6;6,5
12: 6,6


But 2d6 is weighted strongly in the middle, and it tapers evenly to either side.
You could calculate the mean from the above chart by adding up all the pairs and dividing by 36.
You could figure the median on the chart by counting out 18 pairs, and taking the result on either side (they're the same).
And you can see the mode by looking to see which of the lines is the longest.

Lets compare this to a single coin toss. We'll call Heads 1 and Tails 0.

1
0

A coin toss is a flat probability distribution.
The mean is .5, the median is .5, and, most importantly, it has no mode.

So yes, a coin toss will go one way half the time and another way the other half the time.
But 2d6 isn't like a coin toss, it's like a pile of sand.

If you drop a slow stream of sand onto a flat surface, it will make a pyramid with the highest point in the center and the lowest points on the edges. The outcome isn't radically different depending on which side the grain falls, it is actually only a little different, and only a little more different the farther away from the center it lands.

As for being a pessimist, I'd like to point out that I'm often happy that Dice-7=1: every time I use Winterguard to shoot a high armor target. It's not always MY jack taking the hits, after all.

[fire4effekt]

Your mathmatical jargon is nothing but a collection of scribbles from a madman.
Do not listen to these heretics Privateers, I emplore you.
Math and science have no place in war, for it is war that the Gods wage, and we are but simple pawns.
Bow and show allegience to the true Dice Gods lest ye be smithed by eternal misfortune.
Repent for the time is nigh, do not let these blasphamers cloud your judgement anymore.
The Deity are watching, they can see your post count for they lurk on these very forums.

[ringkichard]

I think the confusion here is that the average damage is 1, but the average attack is zero. In your example
only 15 out of 36 attacks did any damage. 58.3% of the attacks dealt no damage.
Probability works better when we are sure everyone is talking about the same thing.

Yes definitely.
And this is where we get, again, to the difference between mean, median, and mode.
The mode attack does no damage. But the mean damage value of an attack is still 1.

[Lee T]

I will have none of your math based chicanary sir! Statistics is a liar and a tool used by liars!

Thankfully it's not statistics, it's probability .

[PG_moz]

It's a long way of explaining that if you are making bets on whether your roll of 2d6 is going to be 7 or lower, it's a smart bet to take (58.33% chance that 2d6-7 does 0 damage). But if you are looking at expected damage from a 2d6 - 7 result, you have to account for the chances that the attack can potentially cause more than 1 point of damage and can never cause less than zero. The average result hovers around 1, which makes it seem more like a coin toss than it is.

[OrsusSmash]

Man, I miss these kinds of threads.

Thanks for reposting all this ringkichard. I was clicking through your sig links the other day, and I was saddened to discover the original link for this one didn't work anymore.

[Chad]

First, this is something that is fairly well known among those that understand the way averages work. The results are skewed simply because of the fact that the would be negatives, those that would make the number an actual average are bumped to 0.

Second, the application of this idea changes drastically with where the zero point exists on the scale. Slide the zero point down, where the POW equals ARM and the seeming average advantage disappears, since all averages exist unaltered by the zeroing effect. Slide it up, where only very high dice rolls and it seems to skew a bit since the 'average' still reports damage (not much) yet there's no way you'll generally bother trying unless you are desperate.

Third, rolls don't happen in a statistically significant series (essentially the same point from the opposite side of the second point). Every single roll is an event into itself. The only math that is really relevant is the odds of that single occurrence. It is the most informative bit of information. In a single roll, if I'm at 2d6 - 7, I cannot reasonably expect a 1+ damage to be my most likely outcome. 58.3% of the time, it will be 0. There is nothing about an average that ever changes these odds on a single dice roll. You should be careful about how you frame this information. Averages are only meaningful with a statistically significant number of rolls. At no point is any 'average' calculation like this one going to bear out the numbers is over the course of thousands, if not hundreds of thousands of rolls.

So the next time your opponent complains that "On average rolls, you shouldn't have done any damage, you lucksack!" just smile at him and know that you're winning through the power of math. Or hit him with a high-velocity tubesock-encased pewter bludgeon... whatever. You're justified.

No you're not. Because you're making a kind of conflation here.

When you speak of averages like this one - you're referring to the statistical result of a skewed (zeroed negatives) POW-ARM calculation. When a player, who is annoyed that your 2d6-8 attacks rolled absurdly high they are referring to an desire for an average dice roll spread, not a statistically significant average calculation. Meaning if you'd rolled an average spread - very high once very low once, the rest in the middle somewhere - then things would have been different. In a way, they are suffering the same problem, in that they cannot predict how a single small series of rolls will go since they can trend extremely high, extremely low and everywhere in between, but that doesn't mean their assertions are wrong, exactly.

People have a tendency to expect (often myself included) that the dice will generally fall some kind of sensible percentages or average, but it rarely works that way for small series of rolls.

[ringkichard]

Your numbers work well for average results when throwing a lot of attacks, but average results are not what you should come to expect from your own models or your opponents attacks! A more effective way to think about the fate of ye old Mangler vs. say 15 Winterguard single-shotting is to sum the damage by probability (important not to incorporate less than zero results!)

Yeah, this is definitely true. What we really want to calculate is the standard deviation of 2d6, and its associated statistical variables.
But look at all the wailing and gnashing of teeth at the probability math in this post. Can you imagine what it would be like if we actually started doing stats?

The results show your numbers are good for guessing what will happen %50 of the time, but it's critical to think about what happens the rest of the time.
[snip]
Better to be surprised at how poorly your opponent does than the alternative.

I hear this a lot, and it's a similar situation to poker, I think.
Some people play very tight, some play more loosely.
Personally, I think a lot of players actually play Warmachine too conservatively (e.g. boosting to hit when they should be buying extra attacks).
Playing conservatively has a cost, and if you're always hedging your bets you're leaving damage on the vine, so to speak.

[themocaw]

Google Docs spreadsheet is here.

Thread with my methodology is here.

Note: I stopped the chart after they reached an average of less than .1 damage, since beyond that, it's vanishingly unlikely that any one member of a full unit would even deal damage.

[ringkichard]

Second, the application of this idea changes drastically with where the zero point exists on the scale. Slide the zero point down, where the POW equals ARM and the seeming average advantage disappears, since all averages exist unaltered by the zeroing effect. Slide it up, where only very high dice rolls and it seems to skew a bit since the 'average' still reports damage (not much) yet there's no way you'll generally bother trying unless you are desperate.

I agree, most people aren't going to bother to use their whole turn of POW 14 attacks against an ARM 24 jack. Dice - 10 is still awful, even if it's not 0.

Third, rolls don't happen in a statistically significant series (essentially the same point from the opposite side of the second point). Every single roll is an event into itself. The only math that is really relevant is the odds of that single occurrence. It is the most informative bit of information. In a single roll, if I'm at 2d6 - 7, I cannot reasonably expect a 1+ damage to be my most likely outcome. 58.3% of the time, it will be 0. There is nothing about an average that ever changes these odds on a single dice roll. You should be careful about how you frame this information. Averages are only meaningful with a statistically significant number of rolls. At no point is any 'average' calculation like this one going to bear out the numbers is over the course of thousands, if not hundreds of thousands of rolls.
No you're not. Because you're making a kind of conflation here.

A couple turns of Winterguard Death Star sprays is potentially 30 or more attacks. You're right that 30 is a low n, and so our confidence interval is going to be fairly wide, but over the course of ten or more games you're going to have an n=300, and at that point your 95% confidence interval is going to be quite tight, less than 5% even.
Even at lower population sizes, what's important here is that people start thinking this way. Yes, if you only have a single Guardsman shooting this isn't a useful heuristic, but infantry comes in units, and it's appropriate to think of attacks in the aggregate.

When you speak of averages like this one - you're referring to the statistical result of a skewed (zeroed negatives) POW-ARM calculation. When a player, who is annoyed that your 2d6-8 attacks rolled absurdly high they are referring to an desire for an average dice roll spread, not a statistically significant average calculation. Meaning if you'd rolled an average spread - very high once very low once, the rest in the middle somewhere - then things would have been different. In a way, they are suffering the same problem, in that they cannot predict how a single small series of rolls will go since they can trend extremely high, extremely low and everywhere in between, but that doesn't mean their assertions are wrong, exactly.

People have a tendency to expect (often myself included) that the dice will generally fall some kind of sensible percentages or average, but it rarely works that way for small series of rolls.

Actually, what I find when people complain about "average dice" (or use "average dice" in theory-machine discussions) is that they really want 2d6 to BE 7, not just average 7. The idea of a statistically average range doesn't occur to some people, because they don't even think of 2d6 as a range, but as an average point. See slokan's response earlier in this thread, for an example.

[Hasten]

Posts like this make me wish fleetingly for a forum "reputation" system. I'm glad we don't have one, but it would be nice to be able to say "Thanks for an awesome post" in a more concrete way. I'll have to make do with:

Thanks for an awesome post!

-H

[ringkichard]

Posts like this make me wish fleetingly for a forum "reputation" system. I'm glad we don't have one, but it would be nice to be able to say "Thanks for an awesome post" in a more concrete way. I'll have to make do with:

Thanks for an awesome post!

-H

You're welcome.
We'll have to make do with the old fashioned reputation system where people do things and then gain or lose reputation based on those actions, just like in the real world.

[solkan]

That would be true if 2d6 was distributed

The distribution of 2D6 is relatively unimportant, what matters is the distribution of 2D6-7 or the answer to the question "How much damage did I take from that?"

I worry about your point when when you bring up the topic of coin tossing and beat me to the explanation of why expected values in situations like this aren't useful. "2D6-7 is 1" is silly for exactly the same reason that "The coin toss is 0.5" is silly.

Let me answer your triangle with the one showing the results for 2D6-7:
00: 1,1;1,2;2,1;1,3;3,1;2,2;1,4;4,1;2,3;3,4;1,5;
5,1;2,4;4,2;3,3;1,6;6,1;2,5;5,2;3,4;4,3
01: 2,6;6,2;3,5;5,3;4,4
02: 3,6;6,3;4,5;5,4
03: 4,6;6,4;5,5
04: 5,6;6,5
05: 6,6

So there are 21 occurrences of 0; and 15 occurrences of 1 or greater.

So see above, for "This is a distribution where it is misleading to talk about expected values and averages."

I expect to take no damage from your attack, but if I take damage, I expect I will take 2 and one third points of damage (on average).

[PG_moz]

I try to dodge getting bogged down in the math by just going straight to programming to handle it. If you're interested in running your own damage probabilities including MAT & POW vs. DEF & ARM, also including boosting and various armor values - I've put together a readily modifiable script on the awesome anydice calculator that does Warmachine/Hordes pretty well.

Fully loaded Ironclad charging
http://anydice.com/program/923

Molik Karn enraged, Last standed, on Zaals feat (spoiler: everything dies)
http://anydice.com/program/925

15 Winterguard vs. our Mangler, and why the example is important (2 man CRAs have higher yield in every probability range)
http://anydice.com/program/927

[Chad]

I agree, most people aren't going to bother to use their whole turn of POW 14 attacks against an ARM 24 jack. Dice - 10 is still awful, even if it's not 0.

Right. There is room between the probability based decision making and the expectation of some kind of average to find a space where most of your decisions are based, more or less, in good understanding of what to expect.

A couple turns of Winterguard Death Star sprays is potentially 30 or more attacks. You're right that 30 is a low n, and so our confidence interval is going to be fairly wide, but over the course of ten or more games you're going to have an n=300, and at that point your 95% confidence interval is going to be quite tight, less than 5% even.
Even at lower population sizes, what's important here is that people start thinking this way. Yes, if you only have a single Guardsman shooting this isn't a useful heuristic, but infantry comes in units, and it's appropriate to think of attacks in the aggregate.

This is the dangers of the aggregate as a decision making foundation - the aggregate isn't the instance, it's only the superset of gathered data. Without mapping the exact results, in precisely the same situations, you cannot know whether your instances, or even, your series of instances will match the average.

Take your example. While the potential attacks is even higher than that, it's fine as a base number because it makes comparison easy.

However, it doesn't truly approach an aggregate because there are too many factors that change the data. First, 12 full winterguard offer a potential of 12 sprays a turn, but it's rare (in my experience) that they actually get this many targets/hits in the same turn. Then there's the problem in types of targets, different ARM values, different number of wounds (single wounders throw off the scale as well, since all damage over one is effectively one), there's misses to consider, casualties in your own unit, etc. In other words, N is really multiple Ns in multiple aggregates of which any single set of rolls is not large enough to make for viable decision making at the time where you're actually making decisions.

Let's say you're facing DEF 13 ARM 17 Cryx Jacks. Now, if you're able to have a full unit survive and have jockeyed for position well enough that 8 of 12 can spray off (you're probably guarding your standard bearer and leader, and you've got a couple of extra held back to take over for the standard bearer should he die), with Gregorovich you're needing 8s on 3 dice so 6.64 hits in a single round. At which point your aggregate begins gathering samples that apply to your sample. If you take any casualties early (a likelihood unless your opponent is ignorant of what those sprays can do) it probably reduces your sample. High or low number of hits affect it as well, as to subsequent casualties, and being engaged, etc., etc. If you're getting 30 landed hits that can roll for damage against the same ARM on multiwound models in a single game, I'd be very surprised. The long and the short of which is this: the point at which N becomes meaningful is not at the interaction, or at the game, or even at the multi-game (like a tournament) level, because your sample of a given combination just isn't large enough.

What can be taken away from this, though, is that breaching ARM, especially around the middle ranges (2d6 -5 to -9) often does more damage on than you might think. Of course, there are no guarantees.

Actually, what I find when people complain about "average dice" (or use "average dice" in theory-machine discussions) is that they really want 2d6 to BE 7, not just average 7. The idea of a statistically average range doesn't occur to some people, because they don't even think of 2d6 as a range, but as an average point. See slokan's response earlier in this thread, for an example.

This might be true some of the time, but I'm not sure everyone thinks of it like that. Since we all play with dice, we (should) all recognize the random factor they are. That doesn't stop us from desiring that, in a sequence of rolls, the probabilities we're trying to make educated guesses about stick somewhat to that percentage.

I've said, many times, "I'd kill for average" which is usually the result of having rolled a disgusting series of low rolls for hit/damage. After the 9th or 10th roll of 5 or below it starts to grind on one's sense of fair play, so we proclaim in the crux of frustration that we'd rather not have high OR low, but more right down the middle, so it can be relied upon. On the opposite end, when my opponent has three rolls in a roll against my ARM 33 Terminus and he not only breaches ARM but kills him with a couple of hits with wildly improbable dice rolls, those same feelings creep up in me, despite the fact that I know it's possible, since it just happened.

It's one of the interesting aspects of Malifaux (another game), you have a set of cards that equate to dice rolls, and you work through the deck. The luck may fluctuate, but at least you know if you've had bad luck in the draw, eventually it will come back around since the good cards are still to come. And vice versa.

[Muffin Man]

Yes definitely.
And this is where we get, again, to the difference between mean, median, and mode.
The mode attack does no damage. But the mean damage value of an attack is still 1.

That's why I always use the term 'expected value'. Since people understand average a certain way it confuses them to redefine what they're thinking about.

This is also relying on the standard deviation to be in your favor though, since the actual expected value is under 1 (ever so slightly). And of course real life results are floored (is that CS-only term? I guess I mean rounded down).

[Jake the Dog]

Bah, you math geeks. I accept every roll as it comes as it is the average roll for me that exact second and that second alone. :P

[Tbunny]

THEORETICALLY the average on 2d6 is 7.


My real-world testing shows it to be 5 or less.

THIS!

T

10Chars

[BryanC]

Super awesome thread! well done ringkichard, bringing the sexy back to Math.

[Nion]

THEORETICALLY the average on 2d6 is 7.


My real-world testing shows it to be 5 or less.

My Stormsmiths would like to submit that the average is actually 10.

[Crazy Uncle Doug]

Though I'd love to argue that my rolling is always bad, that's not entirely accurate. It's more likely that I will notice when my dice fail me, especially when odds are in my favor, than I will when they do as I want. I'm more likely to notice Snake Eyes when I'm trying to roll four or more than I am to notice box cars under the same conditions.

In a given game, dice will generally follow the bell curve, but there will always be outliers in a single given game because 'dice happen', as I say. However, if we were to take all career dice rolls over a Warmachine player's games and graph them out, they'd steer towards that theoretical curve the more games he plays.

'Theoretical' is the key word here. It allows us to make reasonable predictions in a given situation, but no guarantees.

[equilshift]

Bah, you math geeks. I accept every roll as it comes as it is the average roll for me that exact second and that second alone. :P

Might be a good philosophy. The word average is a very loaded word, where the pop-culture definition has all but replaced the mathematical definition (Hint: There is more than one).

The zeroing effect needs to be taken into account by most players (as ringkichard has pointed out) but it is also definitely worth noting that in his dice-7 example, almost 60% of the attacks do 0 damage. I suggest we get some empirical examples and aggregate the data so that we can get RMS and RMSD values, while still being able to see outliers (e.g. that one time where your druids actually killed something ).

[ST&T]

Lets compare this to a single coin toss. We'll call Heads 1 and Tails 0.

1
0

A coin toss is a flat probability distribution.
The mean is .5, the median is .5, and, most importantly, it has no mode.

You forgot 'edge'.

Statistically speaking, it's possible for a coin to fall on edge.

Physics, on the other hand, tell a completely different story.

However, thank you all for posting this for the rest of the forumites to see. It's an excellent tool to remember on the fly in the middle of a game. Not to mention easy to calculate with a variance.

[PG_Pyrodude32]

Nice!

Glad to see this back up!

[Chad]

Statistically speaking, it's possible for a coin to fall on edge.

No, if you assign a numerical value to the heads and tails of a coin, then statistical average will be exactly between those values, since there are only two options.

Math is an abstraction that can apply to real world things, that doesn't mean that every bit of math applies to every real world thing.

There is probability in the results of a coin toss, but no math in toss itself (but math can be applied as an abstraction of the physics involved).

[Hannabull]

I like to think of the dice damage rolls of "what are the odds I'll deal damage?" What are the odds I'll deal damage to that ARM 20 jack with my sentinal? 20-10=10, 10-12=2 or 2/12 or 1/6. So if I shoot that jack with my sentinal, I've got a 1 in 6 chance, or roughly 17% of damaging it. However, if I shoot self same jack with my defender at STR 15: 20-15=5, 5-12=7 or 7/12 or close to 60% chance. If I add an extra dice to it: 5-18=13 or 13/18 or better than 70% chance of doing damage to it. Basically it's a quick and dirty way of thinking the way your maths up there presented everything. The thing is, not only does the chances of you doing damage increase the higher your attack-to-armor ratio gets, the higher your odds of dealing significant damage also increases. Basically the question is this, are you willing to risk a shot for a 1 in 4 chance of dealing damage to the target? How about 50/50. Two-thirds? Three quarters? The point is how low do your odds have to get before you go looking for another target, versus how important is it to take out that target?

[ringkichard]

Right. There is room between the probability based decision making and the expectation of some kind of average to find a space where most of your decisions are based, more or less, in good understanding of what to expect.
This is the dangers of the aggregate as a decision making foundation - the aggregate isn't the instance, it's only the superset of gathered data. Without mapping the exact results, in precisely the same situations, you cannot know whether your instances, or even, your series of instances will match the average.

I'm not sure what you're suggesting here. Throw up our hands and proclaim that probability doesn't work because the instance is never the aggregate, and therefore we can know nothing?
Probability can help us make good decisions, and in this instance probability informs us that our intuition is flawed.
We may expect to weather dice-7 shooting unscathed, minus a few outliers, but the truth of the situation is that even if we are fortunate enough to receive completely average rolls, we should expect the jack to be damaged.
You're absolutely right that we can not expect to always get average rolls, or even to expect average rolls right this moment, but these averages occur for concrete physical reasons governed by the laws of probability.

The only comfort in this situation is the Law of Large Numbers, and reversion to the mean.

[ringkichard]

The distribution of 2D6 is relatively unimportant, what matters is the distribution of 2D6-7 or the answer to the question "How much damage did I take from that?"
I expect to take no damage from your attack, but if I take damage, I expect I will take 2 and one third points of damage (on average).

We agree that the mode Dice-7 attack deals no damage. What concerns me here is the mean.

But I think you're mistaken to expect that you will take no damage from my attack.
If all you know is that I'm going to be making an attack, you should expect to take the expected value of that attack in damage.
There is no other more accurate number.

You're right for things like single wound troopers. Those don't care how much damage they take; a single point might as well be 40. Dead is dead.
The damage you take on average isn't important there, only the % of attacks that will do at least a single point of damage.
That's a situation where the number of misses matters compared to the number of hits.
But for high wound targets, especially warcasters and warjacks, knowing the number of misses and hits just isn't relevant, only the expected damage.