A Mathematician Speaks: Rolling dice

[Kesher]

Lots of great stuff goin' on in this post, especially for those of us who are mathematically challenged!

Vax, in your initial post you mentioned Tunnels & Trolls; if I remember correctly, in T&T, as you beat a monster's "rating" down, its dice+adds for each attack dropped, depending on how much damage had been done to it (I'm gonna have to go look this up...), while the chars remained the same, since they had a different "stat" for damage.  How does this sort of mechanic affect odds?

Also, I'm working on a game right now where resolving conflicts is heavy on the Karma side, with one for sure, possibly two or three d6s thrown as randomizers.  If I were to use as the base die, a 2d6 (instead of 1d6), would it randomize significantly differently than if I used, say, a 1d12?

[Vaxalon]

I wasn't really looking at a round-by-round analysis of TnT; I was just pointing out a major flaw in its design, considering that it encouraged absolutely HUGE throws of dice.

Using 1d12 is VERY different from 2d6.  It will give you extreme values far more often.

You roll a 12 on a D12 once out of 12 times, or 1/12 of the time.

You roll a 12 on a 2D6 once out of 6*6 times, or 1/36 of the time.

[Kesher]

Using 1d12 is VERY different from 2d6.  It will give you extreme values far more often.

You roll a 12 on a D12 once out of 12 times, or 1/12 of the time.

You roll a 12 on a 2D6 once out of 6*6 times, or 1/36 of the time.

That makes a lot of sense, even to an English teacher...  so the design point to consider here is what exactly I'm trying to achieve with the dice rolls in the first place; what's the point of randomizing, in the game-contextual instances of conflict resolution.

I wasn't really looking at a round-by-round analysis of TnT; I was just pointing out a major flaw in its design, considering that it encouraged absolutely HUGE throws of dice.

Oh, yeah, I saw that; I wasn't actually commenting on the point you made, just spinning off of it.  I guess I'm (not very clearly!) wondering how it relates to Noon's observation of DnD:

This means that systems that take account the quality of the hit are double dipping in a way. If your skilled, you will hit more often, which means you will be delivering a better damage average per attack. Now if you do more damage when your roll is beyond the target number required, not only are you hitting more often (with a better damage average delivered per attack), but adding onto the damage from that with more damage.

It's sort of like a death spiral from the other end...if the dudes got more skill than you (attack skill Vs defense skill), your screwed at a geometic rate as the difference increases.

In T&T, as chars wear a monster (or group of monsters) down, the chars stay, if they, say, had 8 dice worth of attacks, at a pretty stable level of result, given the larger number of dice they're throwing.  Results for monsters, however, become more and more "random" as the number of dice in their pool is driven down.  Am I understanding this right?

If I am, I guess I can see how you actually answered my last post before I even posted it :)

Also (this just popped into my head), if Karma as a resolution technique can be described as comparing relative abilities and granting (some level of) victory to the higher score, when does "statistical predictability" (would that be the opposite of randomness?) in the rolling of dice approach "Karmic" levels?


(You know, if I'd had a math teacher who was into gaming, my old report cards would've looked very different...)

[Vaxalon]

Also (this just popped into my head), if Karma as a resolution technique can be described as comparing relative abilities and granting (some level of) victory to the higher score, when does "statistical predictability" (would that be the opposite of randomness?) in the rolling of dice approach "Karmic" levels?

That depends to a great degree on the exact implementation, but as I said, the folks at Steve Jackson Games decided that six dice was as many as they ever wanted to use for any one damage roll.

To refine that idea further;

Let's say that you're up against a weapon that does 1d6x10 damage.  If you want to be reasonably certain of blocking that weapon with GURPS-style DR, then you need to have 60 points of DR.  If you're up against a weapon that does 10D6 damage, you can get away with 45 or 50 points of DR; the chances of getting higher than a 50 on 10 dice are very small.

[Kesher]

Let's say that you're up against a weapon that does 1d6x10 damage.  If you want to be reasonably certain of blocking that weapon with GURPS-style DR, then you need to have 60 points of DR.  If you're up against a weapon that does 10D6 damage, you can get away with 45 or 50 points of DR; the chances of getting higher than a 50 on 10 dice are very small.

Well, okay, so we're seeing it in your example, right?  Fortune-ally or Karma-ically speaking, the weapon can't harm you: Its maximum of 60 pod won't break your DR of 60, only equal it, so you'll always win that conflict; why even roll.

To avoid that clash of Techniques in the example above (& I don't know GURPS very well, so I'm not speaking to that system specifically), design-wise, you might want to consider randomizing the DR as well, yah?  Of course, depending on the ratio of dice rolled (DR might be randomized using d10s for some reason), similar problems might still crop up...

[mindwanders]

Does anyone have a link to a good collection of probability graphs for dice? I just had a quick look and can't really see anything out there.


Alternatively does someone fancy doing some probability graphs for different dice combinations?

I tried to do some as an example (1d12, 2d6, 3d4, 6d2) but I got a bad case of total brain melt on the 3d4's.

[Mike Holmes]

I always post these on math posts: http://www.indie-rpgs.com/viewtopic.php?t=875
http://www.indie-rpgs.com/viewtopic.php?t=5861

To make charts, simply plug the appropriate one into a spreadsheet and go.

Note in the second link, there are some really good further links to John Kim's formulae amongst others. Also, while the first link above has a good presentation of relevant probability formulae, I think one of the last ones is incorrect. Can't remember, it's been a while. Anyhow, if it's really important, get the formula from two sources.

Anyhow, on the subject of "more dice is more predictable" I made the same claim in my rant about opposed rolls. But then Walt pointed out that this isn't neccessarily the case. To a large extent, it depends on what you mean by predictable. Seems obvious at first, but more dice often doesn't meet those definitions. It's best to work with standard distribution and the like if you understand what these are about. The best thing is to understand the actual odds of all of the events, and work from there.


And then, once you have your dice mechanisms set up, playtest them. Because, really, these things can't tell you everything about the feel of the mechanisms used. One common error is using "naturalistic" curves like bell curves to create "realistic" results. Often they don't, and, worse, often realistic is the last thing you want.

Mike

[LordSmerf]

Here are 3 major types of probability curves.  You determine the probablility of a roll by taking the number of unique combinations that give you a given result in X^Y rolls where in 1d6 X=6 and Y=1.

First, flat curves (1d6 in this case), the chance of rolling X in 6 rolls:

1 >
2 >
3 >
4 >
5 >
6 >

Second, bell curves with same sized dice (in this case 2d6), the chance of rolling X in 36 rolls:

2 >
3 >>
4 >>>
5 >>>>
6 >>>>>
7 >>>>>> (notice that the chance of a 7 is the highest)
8 >>>>>
9 >>>>
10 >>>
11 >>
12 >

Third, modified bell curves when rolling multiple, differently sized dice (in this case 1d4+1d6).  Probability here is determined by multiplying the probabilities of the combination together (in this case 4^1*6^1 = 4*6 = 24) so the number of rolls that will be X in 24 rolls is:

2 >
3 >>
4 >>>
5 >>>>
6 >>>> (notice that the chances for 5, 6, and 7 are equal)
7 >>>>
8 >>>
9 >>
10 >

Hopefully this formatting will work, and hopefully some of this will turn out to be useful to someone...

Thomas

[Vaxalon]

To avoid that clash of Techniques in the example above (& I don't know GURPS very well, so I'm not speaking to that system specifically), design-wise, you might want to consider randomizing the DR as well, yah?  Of course, depending on the ratio of dice rolled (DR might be randomized using d10s for some reason), similar problems might still crop up...

That's not how GURPS works, and they're unlikely to change it.  GURPS has a very high level of handling time already; you spend a LOT of time, rolling dice, adding them up, and comparing them to tables and/or other dice rolls, compared to other games, even compared to D20.

[M. J. Young]

I was thinking of posting yesterday to mention that a lot of material concerning how to crunch the probabilities of various dice rolling methods are included in an appendix in the Multiverser Referee's Rules, including bell curves, curves with different dice, and dice pools. There are curves displayed fairly clearly there.

I posted today because

Also, and I don't know if it's appropriate to bring up here, but it would be interesting to note the effect of to hit rolls on damage.

I've got that explained for earlier versions of D&D (those with a closed d20 curve rather than the entirely different open-ended 3E engine) in my page on http://www.mjyoung.net/dungeon/adr.html">ADR's and Surv's. ADR stands for Average Damage per Round, and the page focuses on determining the probable results of N attacks at X probability of success for dY damage, or more simply how a number of attacks, a probability of success, and a range of damage interact, and the impact of altering any one of these.

This means that systems that take account the quality of the hit are double dipping in a way. If your skilled, you will hit more often, which means you will be delivering a better damage average per attack. Now if you do more damage when your roll is beyond the target number required, not only are you hitting more often (with a better damage average delivered per attack), but adding onto the damage from that with more damage.

Which is exactly why Multiverser does it that way--the better your chance to hit, the better your chance to hit well.

Essentially D&D combat is a die pool system where the dice in the pool are compared to a target number (Armor Class) and successes are counted. The only difference between D&D and White Wolf in this regard (other than die size) is that in D&D the dice in the pool are rolled 1 at a time and in White Wolf they're all rolled together.

You just hinted at this the other day, and in principle there's a lot of truth in it, but it's misleading at the same time.

In WoD, the number of dice each side rolls is determined up front. Thus if one side has more dice than the other, that matters immediately. In D&D, though, the number of dice in each pool is determined by the number of rounds the combat ultimately lasts, and the player can make the tactical decision to withdraw or alter his method if it is not going favorably. The ratio of dice in the opposing pools may be fixed (and that's only if neither side changes to an attack mode with a different rate of attack), but the number of dice does not.

That means it is very like a dice pool, particularly (as you intimated in the other thread) in relation to multiple attacks in the round--although even here, if attacks are alternated between sides rather than all one and then all the other, this might not be completely accurate. If I've got two hits to his one, but he can kill me with one hit and I need two to kill him, I might never get that second roll. Yet it is also mechanically distinct from a dice pool in several important ways.

I don't mean to minimize the value of the insight, which I find truly significant, but only to observe that its a similarity that is imperfect.

I also agree that this is something to recognize in design, for exactly the reasons you give.

I hope this helps.

--M. J. Young

[mindwanders]

Also (this just popped into my head), if Karma as a resolution technique can be described as comparing relative abilities and granting (some level of) victory to the higher score, when does "statistical predictability" (would that be the opposite of randomness?) in the rolling of dice approach "Karmic" levels?

It depends how close you want to get. Rolling 100d2 is pretty damn close to giving you a set number every time (around 150 I think, but I'm not going to try and work it out).

However if you fancy looking at a really nice allegedly fate based system that actually opperates a lot like a karma system. You could do worse than look at http://www.fudgerpg.com/fudge/. I've not looked at the actual FUDGE rules, but I read these ones http://www.faterpg.com/ which are based on FUDGE and apparently a little more accessable.

I like the idea of a Karma based system that feels like a Fate based one (so much I've been working on one for a while now). It gives players that familiar "dice in the hands" feeling while not actully affecting play that much.

But if we want to discuss this properly we should probably start a new thread.

[Callan S.]

Ralph, I think your finding a difference between the combat and skill system in D&D because your ignoring the take ten, take twenty and try again rules. The take ten/twenty rules enforce a rigid average without having to roll. While 'try again' rules often means for skills that you can't take 10/20 on means you can just keep rolling to try and get it (though you usually give up resources...but that's the same as combat).

Also it ignores the gang effect. If someone doesn't spot somethingm, for example, it's quite likely one of the three other players will, or atleast everyone's rolls will be a healthy average.

[NN]

Another point to consider is the effect of modifiers. For example, in D&D an innocent "+1 to hit" bonus can have quite different impact in different combats. It has more effect the more unlikely combatants are to hit each other.

[John Kim]

Does anyone have a link to a good collection of probability graphs for dice? I just had a quick look and can't really see anything out there.

Alternatively does someone fancy doing some probability graphs for different dice combinations?

I have a set of essays on dice probability at
http://www.darkshire.net/~jhkim/rpg/systemdesign/

However, I don't have any fancy graphs for these.  Anyone interested in making some visual displays?

[Jay]

In WoD, the number of dice each side rolls is determined up front. Thus if one side has more dice than the other, that matters immediately. In D&D, though, the number of dice in each pool is determined by the number of rounds the combat ultimately lasts, and the player can make the tactical decision to withdraw or alter his method if it is not going favorably. The ratio of dice in the opposing pools may be fixed (and that's only if neither side changes to an attack mode with a different rate of attack), but the number of dice does not.

Hi, just a point of added complexity.. while superficially true (in WoD the number of each side rolls is determined up front).. WoD combat (and resolution of tasks) usually involves MANY rounds... you don't normally roll your dice pool and thats it, combat over.. each roll would usually involve one attack/defense, similar to DnD....

So the aspects of player intervention (retreat etc) are the same.. now where rounds of combat might be more "minimized" in WoD is that generally speaking it is far easier to die or become unconscious in WoD than DnD...

Now.. on another topic.. while I agree that the mathematics are sound.. they only describe the probability.. not the certainty. We have all be in the situation where you had a 99.9% probability of success, and yuet you still failed! ;) Luck still enters into it, despite the mathematics. Similarly, probabilities don't take into account "cruciality".. for example, it is probably "crucial" that when the Red Dragon of Immenent Death (who up until this point has been clawing and batting at me) warms up and decides to blast me with fire, that the Red Dragon fail its roll...

My point is.. probability curves only give you a "sense" of the likely outcome.. but it cannot accurately predict the outcome of combat in which each round might be more or less important the the one before.

Just my 2 or 3 cents ;)

Cheers