The probability and predictability of dice.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Relative range is used by every player every time he sits down whether he realizes it or not. It is used when picking targets. "That brick seems pretty tough he must have at least 25 PD, my attack only averages 20-30 STUN, I think I'll blast the mentalist who only seems to have 15 PD." There, a specifc use of relative range that happens only 30 to 40 times any time any one sits down to play HERO. Dust Raven has said it before and I will say it again:

 

Relative ranges matter before a roll is taken, absolute numbers only matter afterwords.

Both you and Dust Raven seem to have a hard time grasping the difference between "relative" and "absolute". 20-30 STUN is an absolute range. 75%-125% of the mean is a relative range. 25 STUN is a single absolute number. 150% of the mean is a single relative number.

 

Here are some more examples:

12 STUN - absolute

11 or less - absolute

50 or more - absolute

0 BODY - absolute

40-45 STUN - absolute

The maximum possible - relative

the mean - relative

90% of the maximum - relative

less than half of the mean - relative

within 10% of the mean - relative

[PhilFleischmann]

Re: The probability and predictability of dice.

 

You're reading the graph wrong.

Not at all, his graph clearly shows that there is a 16.667% chance of rolling the mean on 2d6, 11.265% of rolling the mean on 4d6, 9.285% chance of rolling the mean on 6d6, 8.094% chance of rolling the mean on 8d6, 7.269% chance on 10d6, etc.

 

Do you see how 16.667 > 11.265 > 9.285 > 8.094 > 7.269 ?

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Phil' date=' why would you dismiss a mathematical fact just because the math isn't used in a game (any game)? We're not talking about a game (any game) here. We're talking about dice.[/quote']

Maybe that's what you're talking about. My claims on this and the other thread have always been about what's relevant to the HERO System. I have never denied the fact that it is less likely for 10 dice to come up all 6's than for two dice to come up both sixes. I have never denied the fact that the percentage variance from the mean will be less the more dice you use. Not only have I not denied these things, I've even asserted them. There are lots of mathematical facts that aren't relevant to the HERO System. If you want to discuss them, I'd suggest that the NGD boards are the place to do it.

 

And there is no such thing as an "absolute range" of results that matters in the Hero System.

Yes there is. If your DEF is 20 and your CON is 20, than any amount of STUN over 40 will stun you. "40 or more" is an absolute range. If a guy is attacking you with 10d6, that's 14.3% or more over the mean, but it isn't the 14.3% that is the significant result, because if the guy is using only 2d6, 14.3% over the mean won't even do any damage at all. Other examples would be Activation Rolls, Skill Rolls, and Attack Rolls. "11 or less," "14 or less," "8 or less" - these are all absolute ranges that matter in the HERO System.

 

You never roll a final result of 10-12 (for example). You roll 10, or you roll 11, or you roll 12. Single results, not a range. On top of that, the purpose of having a range of anything is to compare it to what's not in the range. Ranges are always relative to something else, if the the value of the range is absolute.

Really? What is the range 10-12 relative to? 10 and 12 are absolute numbers that have a specific value. The number 10 doesn't have to be redefined "relative" to something else in order to know how many 10 is.

[prestidigitator]

Re: The probability and predictability of dice.

 

By the way, I realized it might also be helpful to see the Cumulative Distribution Functions of the distributions from post #5. The way to interpret this graph is that, for any x value, the y value of a function represents the probability of rolling less than or equal to that number on the applicable dice.

 

(Click for larger image; again for full size depending on screen resolution)

[Dust Raven]

Re: The probability and predictability of dice.

 

Not at all, his graph clearly shows that there is a 16.667% chance of rolling the mean on 2d6, 11.265% of rolling the mean on 4d6, 9.285% chance of rolling the mean on 6d6, 8.094% chance of rolling the mean on 8d6, 7.269% chance on 10d6, etc.

 

Do you see how 16.667 > 11.265 > 9.285 > 8.094 > 7.269 ?

 

Yep, you're reading the graph wrong. Or just ignoring the facts it presents because you don't believe in them. At this point, I don't care which it is.

[Dust Raven]

Re: The probability and predictability of dice.

 

Maybe that's what you're talking about. My claims on this and the other thread have always been about what's relevant to the HERO System. I have never denied the fact that it is less likely for 10 dice to come up all 6's than for two dice to come up both sixes. I have never denied the fact that the percentage variance from the mean will be less the more dice you use. Not only have I not denied these things' date=' I've even asserted them. There are lots of mathematical facts that aren't relevant to the HERO System. If you want to discuss them, I'd suggest that the NGD boards are the place to do it.[/quote']

Maybe, but the thread was posted here, possibly because of the relavance of probability to any game of change, including the Hero System. In any case, if you're not talking about the same thing the rest of us are talking about, why are you talking?

 

 

Yes there is. If your DEF is 20 and your CON is 20, than any amount of STUN over 40 will stun you. "40 or more" is an absolute range. If a guy is attacking you with 10d6, that's 14.3% or more over the mean, but it isn't the 14.3% that is the significant result, because if the guy is using only 2d6, 14.3% over the mean won't even do any damage at all. Other examples would be Activation Rolls, Skill Rolls, and Attack Rolls. "11 or less," "14 or less," "8 or less" - these are all absolute ranges that matter in the HERO System.

They are ranges, but relative ones, specifically relative to the possible rolls on 3d6 (for the Skill Roll examples) or the possible rolls on damge dice (for the example of rolling enough STUN to Stun someone). But we're just mincing words here, which serves no purpose at all. Let's get back to why you are wrong about dice becoming less predictable to more you roll them.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Yep' date=' you're reading the graph wrong.[/quote']

No, I'm reading the graph correctly, but I'll bite: What do you see the graph showing as the probability of rolling the mean on 2d6, 4d6, etc.? Something other than the figures I listed? Do you see the tallest, pointiest probability curve? That's the one for 2d6. Do you see how it reaches higher than the other curves? Do you see the curve with the lowest hump in the middle? That's the one for 10d6. Do you see how the height of the points on the curve display the probability of rolling the particular number? Do you see how lower points represent lower probability? Or is 7.3% greater than 16.7% in your world?

 

Maybe, but the thread was posted here, possibly because of the relavance of probability to any game of change, including the Hero System. In any case, if you're not talking about the same thing the rest of us are talking about, why are you talking?

By "the rest of us" you mean you. Many of us are talking about things that are relevent to HERO. There are many ways to look at the statistics of rolling dice. Some of them are irrelevent to the HERO System. If I've agreed with you on statements about some of them (which I have), then why are you still arguing?

 

They are ranges, but relative ones, specifically relative to the possible rolls on 3d6 (for the Skill Roll examples) or the possible rolls on damge dice (for the example of rolling enough STUN to Stun someone).

Again, you demonstrate a complete lack of understanding of the meaning of the words "relative" and "absolute". I take it that to your mind, there is no such thing as an absolute range. If I'm wrong, could you give me an example of what you think is an absolute range?

 

But we're just mincing words here, which serves no purpose at all.

Neither of us is mincing words. I'm using words correctly, and defining them for those who don't know what they mean. You are misunderstanding words, no matter how many times I try to explain them. If you don't understand the meaning of the words I am using, then we may as well be speaking two different languages, which indeed would serve no purpose at all.

 

Let's get back to why you are wrong about dice becoming less predictable to more you roll them.

"Back to"? When did we even start? If we define predictable as "having a lower expected *relative* variance", then yes, rolling more dice is more predictable. I've said this many times now, in many different ways. We seem to be in agreement on this point.

[prestidigitator]

Re: The probability and predictability of dice.

 

Okay. It seems I have overcome my laziness (mostly due to impatience over the conversation ). Here is a table of average damage vs. DEF. The "DEF-mean" value should be taken to be the difference between the DEF against which the attack is being applied and the mean of the roll under who's column you are looking (this is equivalent to adding fixed values to each roll such that the mean die rolls become the same; so it is like applying the attacks 2d6+28, 4d6+21, 6d6+14, 8d6+7 and 10d6 all against a common DEF of 35+"DEF-mean"). This will apply to the Stun of Normal Attacks vs. DEF or the Body of Killing Attacks vs. rDEF (i.e. the count of the actual pips applied directly against some kind of defense value; it is not in any way applicable to the Body Count of Normal Attacks or the Stun of Killing Attacks). The values are the mean damages that will get past the DEFs for that number of dice.

                      Mean Damage
          --------------------------------------
DEF-Mean    2d6     4d6     6d6     8d6     10d6
--------   ------  ------  ------  ------  ------
-10        10.000  10.000  10.007  10.028  10.061
-9         9.000   9.001   9.017   9.052   9.100
-8         8.000   8.005   8.037   8.090   8.158
-7         7.000   7.016   7.073   7.151   7.242
-6         6.000   6.043   6.133   6.242   6.358
-5         5.000   5.097   5.230   5.372   5.514
-4         4.028   4.194   4.374   4.551   4.719
-3         3.111   3.353   3.580   3.788   3.980
-2         2.278   2.593   2.860   3.094   3.304
-1         1.556   1.928   2.223   2.474   2.696
 0         0.972   1.372   1.676   1.933   2.160
 1         0.556   0.928   1.223   1.474   1.696
 2         0.278   0.593   0.860   1.094   1.304
 3         0.111   0.353   0.580   0.788   0.980
 4         0.028   0.194   0.374   0.551   0.719
 5         0.000   0.097   0.230   0.372   0.514
 6         0.000   0.043   0.133   0.242   0.358
 7         0.000   0.016   0.073   0.151   0.242
 8         0.000   0.005   0.037   0.090   0.158
 9         0.000   0.001   0.017   0.052   0.100
10         0.000   0.000   0.007   0.028   0.061

And here is a graph of the same:

 

(Click to enlarge; again for full size depending on screen resolution)

[Dust Raven]

Re: The probability and predictability of dice.

 

"Back to"? When did we even start? If we define predictable as "having a lower expected *relative* variance", then yes, rolling more dice is more predictable. I've said this many times now, in many different ways. We seem to be in agreement on this point.

Heh. Hahahahahaha.

 

Funny this is the first time you've said you agree without turning right around and stating the opposite immediately after. In any case, not having the same conversation I've been having could explain a lot.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Funny this is the first time you've said you agree without turning right around and stating the opposite immediately after. In any case' date=' not having the same conversation I've been having could explain a lot.[/quote']

I can understand how you might think that if you don't understand the words I've been saying. If I say, "I like vanilla ice cream," and then I say, "I don't like chocolate ice cream," and you don't understand the meaning of, or the difference between the words "vanilla" and "chocolate," it might seem to you that I am saying one thing and then immediately saying the opposite. If all you can understand is "I like ice cream" and "I don't like ice cream," then it would certainly seem like I am making two contradictory statements.

 

On the subject of dice probability, I have only ever made three different statements. All the rest have been either clarifications of those statements, or logical arguments in support of those statements. The three statements are:

 

1) If we define predictable as "having a lower expected *relative* variance", then the more dice you roll, the more predictable the results.

 

2) If we define predictable as "having a lower expected *absolute* variance", then the more dice you roll, the less predictable the results.

 

3) It is the absolute variance, not the relative variance, that matters in the HERO System.

 

We all seem to be in agreement on statement 1. Statements 1 and 2 are not contradictory, and can only be seen as such if you ignore the words "relative" and "absolte".

 

To me, all three of these statements are obvious matters of fact. None of them are subjective matters of opinion. It seems to me that anyone who disagrees with these statements either doesn't understand them or is using faulty reasoning. I don't mean to be insulting. It's observable, demonstratable, mathematical fact. If one person says 2+2=4, and another says 2+2=5, then one of them is wrong.

 

And the only arguments I've seen against statements 2 and 3 are based on not understanding the meanings of "absolute" and "relative."

[SteveZilla]

Re: The probability and predictability of dice.

 

A roll of 2d6 has a standard deviation of 2.42 and a mean of 7.0. 68.2% of the time the roll will be within 7.0 + or - 34.5%.

 

I'm confused. Plus or minus 34.5% of what?

 

A roll of 10d6 has a standard deviation of 5.40 and a mean of 35.0. 68.2% of the time the roll will be within 35.0 + or - 15.4%.

 

And again. Plus or minus 15.4% of what?

 

Edit: removed extra php code

 

What PHP code was that? I'd like to see it!

[Dust Raven]

Re: The probability and predictability of dice.

 

On the subject of dice probability, I have only ever made three different statements. All the rest have been either clarifications of those statements, or logical arguments in support of those statements. The three statements are:

 

1) If we define predictable as "having a lower expected *relative* variance", then the more dice you roll, the more predictable the results.

 

2) If we define predictable as "having a lower expected *absolute* variance", then the more dice you roll, the less predictable the results.

 

3) It is the absolute variance, not the relative variance, that matters in the HERO System.

 

Maybe it's true that I don't know what you mean then. As far as prediting the likely results of die rulls, the statement you make in #2 is irrelevant. It seems to me that what are saying in #2 is that it is more difficult to predict any single possible result (or a group of the same number of results) the more dice you roll. That doesn't matter though, and is not how I've ever definded predictability in the context of rolling dice.

 

#3 is just blantantly wrong. Why you consider it a fact of any kind is beyond me.

 

#1 is the only thing we can seem to agree on, except you seem to think it doesn't matter when playing a game using dice. It makes me wonder when you do think it matters.

 

 

I have another game example if you are interested (you seem to have ignored my previous one despite your tenacity in asserting that the predictability of dice can only be measured when playing a game).

 

A character has two attack powers:

 

RKA 3d6+1

EB 10d6

 

Neither attack does BODY and the target has a DEF of 15 and a CON of 20. Which attack can you more accurately predict the results of and which should the character use against his target to cause the most possible damage over several attacks?

 

I believe that the 10d6 EB is more predictable. You roll more dice than compared to the RKA (where you roll 4, 3 for the BODY damage an one more to determine the STUN). Now with the target's DEF+CON=35, and 35 being the mean/average/middle possibility of either attack, it matters not which I use for purposes of Stunning; either have a roughly 50/50 chance of this. For overall damage done over several attack though, I feel the RKA is the best choice because is it more random. While you are equally likely to roll really low (so low as to do no damage) as you are to roll really high, the extra damage that bypases the target's DEF on those high damage rolls will occur more often than the more predictable EB and cause more damage in a sustained combat.

 

Does your opinion on what measures predictability disagree with this assessment? If so, how?

 

P.S.: I'm kina hoping I've finally gotten where you are coming from on this, especially considering how you keep mentioning how all this only matters while playing a game. I think you are coming at this from the final results end (saying the KA is more predictable because you can predict it will cause more damage) and we just aren't quite meating in the middle because I'm coming at this from a purely mathematical standpoint that doesn't change whether you are playing a game or not.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Maybe it's true that I don't know what you mean then. As far as prediting the likely results of die rulls' date=' the statement you make in #2 is irrelevant. It seems to me that what are saying in #2 is that it is more difficult to predict any single possible result (or a group of the same number of results) the more dice you roll. That doesn't matter though, and is not how I've ever definded predictability in the context of rolling dice.[/quote']

Ah. It seems that indeed you don't know what I mean. What you say here is not what I mean by predictability at all. Let me try to rephrase the definition of predictability as I've been using it: Predictability is the narrowness of the range of likely results, or how far will a result be likely to vary from the mean? The farther from the mean a result is likely to vary, the less predictable it is. This can be reflected quantitatively in the standard deviation. If you like, you could think of predictability as the inverse of the standard deviation - the higher the SD, the less predictable; the lower the SD, the more predictable. It can also be seen as the likelihood of results near the mean. The five results nearest the mean on 2d6 have probabilities ranging from 11.11% to 16.67%. The five results nearest the mean on 10d6 have probabilities ranging from 6.82% yo 7.27%. The lower the probability of results near the middle, the less predictable.

 

#3 is just blantantly wrong. Why you consider it a fact of any kind is beyond me.

It's because you don't understand what I mean by "absolute" and apparently also by "predictability." No one has provided any example of an instance in which "75% of the mean" or "80% of the maximum" are useful statistics in HERO.

 

#1 is the only thing we can seem to agree on, except you seem to think it doesn't matter when playing a game using dice. It makes me wonder when you do think it matters.

I'd be happy to hear of any game where it does matter. In general, I'd say that relative variance rarely matters as much as absolute variance. Though I'm sure there are some RW applications where it matters, say in political forecasts, disease and health statistics, maybe weather.

 

I have another game example if you are interested (you seem to have ignored my previous one despite your tenacity in asserting that the predictability of dice can only be measured when playing a game).

I don't remember what your other example was that you say I ignored. I'll try to go back and find it later. And I never said that predictability of dice can only be measured when playing a game - another example of you not understanding me. I said there are many different ways to look at the statistics of dice rolls, some of which are relevent to games and some are not.

 

A character has two attack powers:

RKA 3d6+1

EB 10d6

This is a completely different example than anything I've been talking about up to this point. But that's OK. We can still compare the predictabilities of these two rolls. Up until now, I've generally been talking about comparing things like 10d6 to 2d6+28, and I say the latter is more predictable in absolute terms. Here, we're comparing 10d6 to (3d6+1)x(1d6-1). And yes, the latter is definitely less predictable. That is to be expected when you multiply a random number by another random number. With the killing attack, the expected absolute variance is much higher. The standard deviation on 3d6 (+1) is a little less than 3. The standard deviation on 1d6 (-1) is about 1.7. Therefore the SD for the STUN of this KA is about 5.1. Hmmm... the SD of the 10d6 is a little over 5.3, so it seems 10d6 is still less predictable. That doesn't seem right. What did I forget? Oh! The +1 on the 3d6 is being multiplied by the STUNx, and therefore needs to be taken into consideration. So the SD of the KA is really more like 4 x 1.7 = 6.8. That's more like it! It would have been simpler to compare 9d6 N with 3d6 K. The SD's are just a tad over 5 and over 5.1 respectively. (I also didn't take into account the minimum of 1 on the STUNx, but I wanted to keep it simple, and it doesn't have that much effect on predictability.)

 

P.S.: I'm kina hoping I've finally gotten where you are coming from on this,

Me too.

especially considering how you keep mentioning how all this only matters while playing a game.

I've never said that. So much for getting where I'm coming from.

I think you are coming at this from the final results end (saying the KA is more predictable because you can predict it will cause more damage)

No, that isn't what I'm saying at all.

and we just aren't quite meating in the middle because I'm coming at this from a purely mathematical standpoint that doesn't change whether you are playing a game or not.

I'm also coming at this from a purely mathematical standpoint. Some math is useful in the game and some isn't.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

I assume this is the example you mentioned that I ignored:

 

Deduction Dan has two Attack Powers:

 

EB 12d6, STUN Only (vs. ED)

EB 4d6, AVLD (Power Defense)

 

DD knows his target has an ED of 54 and Power Defense of 18. (I picked these numbers based off of 75% the total damage possible for the attack they defend against to represent a stastical equivilant for each.)

 

Which attack should Deduction Dan use?

And while this is certainly a valid and relevent question to ask in the HERO System, it really doesn't have anything to do with what I was talking about in regard to predictability of the dice rolls. DD can use whichever attack he wants. The concept I was getting at was which dice roll is more predictable. And as I've said before, more dice = less predictable. 4d6 produces a range of 4-24 - 21 possible results, ranging in probability from 11.265% to 0.077%. 12d6 prodcues a range of 12-72 - 61 possible results, ranging in probability from 6.654% to 0.0000000459%. The most likely result on 4d6 is almost 11.2% more likely than the least likely result. The most likely result on 12d6 is only 6.654% more likely than the least likely result.

 

How's that for another example of what I mean by predictability?

 

Back on the original thread I mentioned other ways of using fewer dice (for an equivalent DC attack), and how they would change the predictability. Some methods can decrease the predcitability, like using 4d6 x 4 instead of 12d6. When you multiply fewer dice by a factor to bring it up to the same mean, the variances are all multiplied by the same number, creating a much wider variance. The standard deviation is multiplied by the same factor. Whereas using fewer dice and adding a fixed number to bring it up to the same mean doesn't change the variance or standard deviation of those fewer dice at all. Thus:

 

4d6+28 is more predictable* than 12d6, and

4d6 x 4 is less predictable* than 12d6.

 

*And again, I mean in absolute, not relative terms.

 

A long time ago, there was a thread (more than one IIRC) about changing the 3d6 bell curve for attack and skill rolls. I mentioned a few possible methods. To better understand each other, here is what I say about some of them, based on the definition of predictability that I've been using:

 

d4+d6+d8 - a slightly less predictable bell curve than 3d6

5d4-2 - a significantly more predictable bell curve than 3d6

d16+2 - (assuming you can contrive a d16**) a flat "curve" where all results are equally likely - far less predictable.

 

Would you agree with these?

 

**You actually could simulate a d16 by using 2 d4's: one numbered 1-4, and the other numbered 0, 4, 8, 12. Or using a d2 (a coin) and a d8.

[Dust Raven]

Re: The probability and predictability of dice.

 

Okay, Phil. I'm now convinced that you just don't know what you are talking about. You should look up information about statistics in general and specifically review the Law of Large Numbers. Right now you are just fiddling with numbers and thinking your results make sense. I have no idea why you think your results make sense (myself and others have shown you proof you are doing the math wrong), but to you they do and that's apparently enough to keep you happy.

[prestidigitator]

Re: The probability and predictability of dice.

 

He makes plenty of sense to me. I think you are just continuing to talk past each other without really being able to see eye-to-eye for some reason, despite the obvious attempts.

[Dust Raven]

Re: The probability and predictability of dice.

 

He makes plenty of sense to me. I think you are just continuing to talk past each other without really being able to see eye-to-eye for some reason' date=' despite the obvious attempts.[/quote']

 

Then maybe you can explain it to me.

[prestidigitator]

Re: The probability and predictability of dice.

 

Then maybe you can explain it to me.

Maybe, but don't count on it.

 

How's this: look at the results we are after. What does it mean ( ) to you, in a real game, if I say someone hit me and managed to do 75% of their average damage to me? Nothing. Statistics deals with real and measurable results, and the relative amount of a roll just isn't the metric that makes sense given that we mark down measurable units (Stun and Body) of damage on our character sheets. Likewise--with the exception of Damage Reduction--Defense Powers don't work vs. a relative amount; they block a set number of units (Stun and Body) of damage. Does that help any?

[Anatosuchus]

Re: The probability and predictability of dice.

 

What does it mean ( ) to you' date=' in a real game, if I say someone hit me and managed to do 75% of their average damage to me? Nothing. Statistics deals with real and measurable results, and the relative amount of a roll just isn't the metric that makes sense given that we mark down measurable [i']units[/i] (Stun and Body) of damage on our character sheets. Likewise--with the exception of Damage Reduction--Defense Powers don't work vs. a relative amount; they block a set number of units (Stun and Body) of damage. Does that help any?

This, however, has nothing to do with predicting the amount of damage you are likely to do to a foe with a given die pool. Which is the point of the thread.

[Dust Raven]

Re: The probability and predictability of dice.

 

Maybe, but don't count on it.

 

How's this: look at the results we are after. What does it mean ( ) to you, in a real game, if I say someone hit me and managed to do 75% of their average damage to me? Nothing. Statistics deals with real and measurable results, and the relative amount of a roll just isn't the metric that makes sense given that we mark down measurable units (Stun and Body) of damage on our character sheets. Likewise--with the exception of Damage Reduction--Defense Powers don't work vs. a relative amount; they block a set number of units (Stun and Body) of damage. Does that help any?

 

I've never argued about the terminology used in the game, only that the mathematical prediction of any given role (and the math/terms used to determine it) doesn't just "go away." In other words, the reality of what is and is not predictable does not change just because we aren't using the terms to desribe it (or it's comparison to the actual results) while playing.

 

In other, other words, if 75% of my average damage is 32 points, saying "the attack had 32 points of effect" in game does not change the fact that 32 is still 75% of the average damage. This is a wacked out example though, as it is describing the results, not predicting them.

 

My point has generally been, that when discussion predictability, these facts should not be ignored even though we don't typically use them when describing the final result of any given roll.

[prestidigitator]

Re: The probability and predictability of dice.

 

Ah, but statistics is all about the results. It is the results you have to look at when considering your distribution. We can talk about flat, normal, and composite distributions all we want, but they mean nothing until we assign them a real scale that describes our results.

 

Let me present you with an example. Let's say for now that we are going to roll 3d6. First our damage is 3 DCs of Normal Damage and we are targetting someone with a 10 DEF (I'm going to deal in Stun only). Cool. We know we have a little over 50% chance of damaging them, and we know most results are going to fit within ±3 of 10. If we roll a 13 on the dice, we are about one standard deviation above the mean (also about 30% above it). The target takes 3 Stun.

 

Now what if we roll those same 3d6, but we roll it for a 6 DC attack and multiply the physical number of pips in the dice roll by two to get the points of effect. Now the mean is 21, and most of the results are going to fit within ±6 of 21. The chances of doing damage to the target are very good; in fact there's only a very slight chance we won't do damage. If we roll a 13 on the dice, that is 26 damage. We are still about one standard deviation (30%) above the mean, but our results are dramatically different! Even if we increased the target's DEF to 21 (where there would still be about a 50% chance of damaging them), I think you can probably see the results will be different: when we do roll above the average, it's likely to be in the 21-27 range, not the 21-24 range.

 

This isn't quite so dramatic when we increase the number of dice instead of multiplying the roll by a constant to increase the DCs, because the standard deviation is proportional to the square root of the number of dice instead of going up linearly with the number of DCs. BUT the positive correlation is still there: when you increase the number of dice, both the mean and the standard deviation do increase.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Okay' date=' Phil. I'm now convinced that you just don't know what you are talking about. You should look up information about statistics in general and specifically review the Law of Large Numbers. Right now you are just fiddling with numbers and thinking your results make sense. I have no idea why you think your results make sense (myself and others have shown you [i']proof[/i] you are doing the math wrong), but to you they do and that's apparently enough to keep you happy.

Well, since you've ignored all of my questions, I don't have much choice but to draw the same conclusion about you. I don't want to do that because I prefer to think that when someone disagrees with me on an objective mathematical fact, it is probably a matter of misunderstanding, lack of clarity, we're using different words to say the same thing, or we're using the same words to say different things. Maybe I'm just more charitable in this way than you.

 

Fine, so we each think the other doesn't know what he's talking about. Unfortunately, that doesn't get us anywhere. I don't mind disagreeing, but I'd at least like to be clear on what the source of that disagrement is. I prefer clarity to agreement. But I guess it is not to be in this case.

 

I have never denied, or made any attempt to refute the Law of Large Numbers. I'll review it if you like, but while I'm doing so, "you should" calculate the probability of rolling 35 on 10d6 yourself. Then calculate the probability of rolling 35 on 2d6+28. (The answers I get are 7.269% and 16.667%, respectively.) Then ask yourself which is the higher probability. Then "you should" calculate the probability of rolling in the range of 34-36 on 10d6 and on 2d6+28. (My answers are 21.575% and 44.444%.) Which is higher? Then try the range 30-40. (I get 68.697% for the 10d6, and 100% for the 2d6+28.)

 

If you think that you or anyone else on this thread or the other thread have "prooven" that I'm doing the math wrong, then you obviously don't know the meaning of either the word "proof," or the word "math," or the word "wrong," or more than one of these.

 

I get the feeling that you have some emotional stake in this argument. I have no idea why, but it seems to be impairing either your writing clarity or your reading comprehension, or both.

 

Like I said on the other thread, I remember having intelligent conversations with you on the HERO boards before, and I believe and hope we will do so again in the future. I don't know what went wrong this time on this particular subject.

[schir1964]

Re: The probability and predictability of dice.

 

Well I can now post and say that I totally misunderstood something completely about PhilFleischmann's comparisons that he was making.

 

I presumed he was trying to compare two relatively similar dice rolls, when in fact, he isn't. So I really don't see how any sort of relevant conclusion can be made concerning predictability.

 

10d6 vs 2d6+28

 

These two dice rolls don't have enough in common to come to any conclusion about thier predictablity.

 

10d6 has an absolute range of 10 to 60 for it's domain.

2d6+28 has an absolute range of 30 to 40 for it's domain.

 

So as far as I'm concerned, thes two dice rolls aren't even comparable for predictability since the ranges aren't the same. If the two rolls had the same range for comparison and one had higher percentages for the middle, then I would be interested in the method used to obtain that result.

 

- Christopher Mullins

[SteveZilla]

Re: The probability and predictability of dice.

 

A character has two attack powers:

 

RKA 3d6+1

EB 10d6

 

Neither attack does BODY and the target has a DEF of 15 and a CON of 20. Which attack can you more accurately predict the results of and which should the character use against his target to cause the most possible damage over several attacks?

 

I believe that the 10d6 EB is more predictable. You roll more dice than compared to the RKA (where you roll 4, 3 for the BODY damage an one more to determine the STUN). Now with the target's DEF+CON=35, and 35 being the mean/average/middle possibility of either attack, it matters not which I use for purposes of Stunning; either have a roughly 50/50 chance of this.

 

Actually, the RKA 3d6+1 has a mean Stun result of 30.667. An RKA doesn't produce the same range and mean as an EB of the same cost, in part because of the -1 on the stun multiple, and in part because it's a multiple. Multiplying a dice probability graph by a number doesn't produce a graph identical to a larger group of dice.

 

I will be posting a graph of the potential Stun damage of a 3D6+1 RKA soon.

[Dust Raven]

Re: The probability and predictability of dice.

 

I have never denied, or made any attempt to refute the Law of Large Numbers. I'll review it if you like, but while I'm doing so, "you should" calculate the probability of rolling 35 on 10d6 yourself. Then calculate the probability of rolling 35 on 2d6+28. (The answers I get are 7.269% and 16.667%, respectively.) Then ask yourself which is the higher probability. Then "you should" calculate the probability of rolling in the range of 34-36 on 10d6 and on 2d6+28. (My answers are 21.575% and 44.444%.) Which is higher? Then try the range 30-40. (I get 68.697% for the 10d6, and 100% for the 2d6+28.)

This is not an example of the Law of Large Numbers. This is not an example of less dice vs more dice. Is that clear? You wanted clarity.

 

If you think that you or anyone else on this thread or the other thread have "prooven" that I'm doing the math wrong, then you obviously don't know the meaning of either the word "proof," or the word "math," or the word "wrong," or more than one of these.

I take this an an insult and an end to our discussion.