Okay, this is me trying to be helpful and offering what I know. I tried to see if it was covered before, elsewhere, and I don't see it, but I could easily be wrong. Also, for some of you, this will be "Well, duh" but I have found that some people can be remarkably math-phobic, even in the gaming community. So here's what I got, edited down for the mathophobe.

One of the questions asked of mathematicians is, "If the most likely outcome of an event is X, how far do you have to go from X before you have encompassed most of the possible events?"

This is a question gamers want the answer to. Let's say you're playing old-fashioned way-back Tunnels and Trolls. You and your party are getting seven dice when you fight, but the troll is getting eight. How often do you roll higher than he does, so you can do damage to him?

This is where the standard deviation comes in.

One standard deviation from the mean (that's the fancy mathematician word for average) in both directions covers roughly 68% of the results. Two standard deviations covers 95%, and three standard deviations covers 98%.

I'm not going to bore you with how to calculate standard deviations, that's not really useful to the game designer. The point I'm going to make here, is that the more dice you use, the LESS random the result is! That may sound counterintuitive, but it's quite true. This is true whether you're summing the dice, averaging, taking the best, counting the 10's, whatever.

How many is too many? That depends on your game, and the mechanics you derive from the dice, but it should be noted that the folks over at Steve Jackson games decided that anything over six dice just didn't have enough variation. Above that score, you roll four, five, or six dice, and multiply by some factor. 60 dice isn't rolled; instead you roll 6d6 and multiply by ten.

And now we'll get to the point. When creating a resolution mechanic, you need to consider the variability of the dice you're using. A roll of a single die is much more likely to create "wild" results than a roll of several dice.

Hi Fred,

More good topics.

I've fully with you on the discussion of distributions and probabilities, and you'll find another enthusiastic participant in Mike Holmes.

If I'm not mistaken, what you're calling attention to is the difference between a flat and bell (or other curve) distribution. My favorite gaming example is, indeed, D&D of any period. You're rolling a single die with quite a range: 5-100%, broken into 5% increments.

Now, despite horrible eye-rolling and teeth-gnashing to the contrary, we all really know that there is an equal chance to roll a 20 as a 3, or any other single value, on a single die. The difference between a 2 or less and an 18 or less cannot be observed if each character is given only one chance to hit. It can only be observed across multiple instances of trying to hit (in which case the guy with 18 is better, duh).

But here you are, with a low-to-mid level D&D character. The quick way to get an effective character is not by increasing one's chance to hit - that comes too slowwww! And it really isn't going to make a difference in those crucial situations when you have 4 hit points left and simply must kill the troll right this turn. No, what you needed was a +12 Monster-Dicer sword, not for its increased chance to hit, but mainly for that monstrous damage. As you essentially "swung wild" per hit, making each one count is the key to success.

Same goes for the real meat in any rules-set which relies on a flat line distribution: getting multiple attacks. When you get those three attacks in one turn, in D&D, watch out! Because effectively, you didn't just get three attacks - you got one attack, using a 3d20 dice pool, with your damage dice multiplied by the number of successful dice in the roll.

Fred, is this along the lines of the issues you wanted to discuss in this thread? I'll do some hunting for the discussions from a couple years ago about sinusoid curves (bell curve offense vs. bell curve defense).

Best,
Ron

A good point Vaxalon. Though it was clear to me already, I feel that many folks propably hadn't thought about it. This is a classic karma / fortune issue. People designing, tinkering with and playing games need to be aware of the choices they are making.

Many of the games here at Forge seem to be dice pool based games. When characters get more competent, fortune doesn't effect them as much.

One thing that I like about HeroQuest, is that you can pretty much tweak your drama/karma/fortune balance by yourself.

EDIT: Ron, an interresting point you make. D&D does have a bell curve, because each situation really needs several rounds of action, possibly with multiple attacks or something per round. On the other hand, HeroQuest has truly has a flatline, because a conflict is a conflict. Well, at least in simple contest that is. :) My players are all too aware that it's much harder to beat the odds in an extented contest. =)

Hiya,

Newsalor, that's right: consequence is also an issue. Most players in HeroQuest, when faced with a conflict which might include character death or anything else that's pretty undesirable, will shoot for an Extended Contest. In that case, the degree of consequences of single rolls are determined by bidding points, modified to be sure by the dice, but not wholly dictated by them.

One of the frustrating elements of low-level D&D play is that one's entire "imaginative entry into play" (the character) is at often risk based on the outcome of a single flat-curve roll.

Best,
Ron

Ron Edwards wrote: If I'm not mistaken, what you're calling attention to is the difference between a flat and bell (or other curve) distribution.

One thing that i find interesting is the shape of a probability curve generated by unmatched dice. Rolling 2d6 gives you a triangle with its highest point found at the value 7. Rolling 1d6+1d4 results in a triangle with a flat top centered on 6.

I am not sure that this is direcly relevant to the discussion at hand, but it is a rarely used option for probability curves...

Thomas

Heh. It's not. But you're right that it's an option that creates subtle differences in probability distribution that people should be aware of.

Vaxalon wrote: Yes, but it's not just that; it's also that the MORE dice you're using, the more "spiky" the bell curve gets; the more dice you roll, the closer the results cluster around the average.

I'm going to make here, is that the more dice you use, the LESS random the result is! That may sound counterintuitive, but it's quite true.

Hiya,

Fred, the point about more dice yielding a more consistent average is clear. It directly relates to my own concern in the game Over the Edge, in which the crucial shift from 2d6 (most people at most things, and characters at most things) to 4d6 or 5d6 plays, in my opinion, an overly-intrusive role in the system.

I'm not sure if you're familiar with multiple-dice systems that do not rely on adding up the dice, or for that matter, even target numbers. The basic parent of these systems is probably Prince Valiant, which uses d2 (coins), but its prolific spawn include Story Engine, Sorcer, InSpectres, and many others.

Or conversely, consider "add-up" systems which do not rely on quantity but rather rank. I think the ultimate rock-solid "one curve to rule them all" game is Fudge, which relies on d3's (actually d6 with two plusses, two minuses, and two blanks). You roll four of them and superimpose the results on a seven-step ladder of competence (mediocre, fair, good, etc), starting from the point your character's skill or whatever is rated at.

The point of Fudge is to make sure that all characters are subject to the same standard deviation no matter what their basic competence is, and it it succeeds very well. Whether this is a good design consideration is a matter of local judgment and priorities.

Thomas, one game I've always wanted to play in order to examine the dynamics of the point you're making is Alternity. Wasn't there a thread or two about Alternity in Actual Play ...? Gotta check.

GBSteve, yeah! One of the things I like about HeroQuest is that bidding is literally the degree of investment, which also means the degree of risk, which also means the degree of unpredictability. I was very influenced by the dynamics of HeroQuest in designing Trollbabe.

Best,
Ron

I'm familiar with OTHER systems that don't rely on adding up dice, such as WoD. The thing is, it doesn't really matter what you're doing with them; adding them up, counting the number that come up with a particular value or more, etc., the fact remains that the more dice you use, the less variability you end up with.

Fudge has taken a bold step by always rolling what amounts to 4d3-8 to get a range from -4 to +4, clustered around zero. I personally don't think it's a good design consideration, because some endeavors have an inherently greater amount of "chaos" in the outcome.

I've never played Alternity.

Understanding probability curves and dice pools is a very powerful tool for game designers. Understanding the implications of what seems to be a straight forward choice can make all the difference.

For instance, few people would consider D&D (and we'll talk 3E here though its true of all editions) a die pool system. After all you roll a single d20 with a linear curve right?

Not in combat you don't. No combat in D&D (save at the lowest levels) is resolved with 1d20 roll. Its resolved only after a series of d20 rolls, the number of which is set by HitPoints/average damage per roll suffered (where per roll counts misses as 0 damage in the average).

If a combat lasts 8 rounds, than in terms of probabilities its the equivelent of rolling 8d20 (before accounting for multiple attacks).

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.


What causes a major breakage in the d20 system is that ONLY combat is handled this way. Everything else you might want to do in the system is handled by a single d20 vs Difficulty Class...usually with just 1 single roll.

So what you have is a normally distributed (mostly) bell curve for combat resolution, paired with a linearly flat curve for everything else. This is not good for many reasons, not the least of which is the different scale by which Difficulty must be measured on. Having a 50% chance to succeed on a single d20 is not that bad when you'll be rolling 8-10 d20s over the course of a battle. Having a 50% chance to succeed on a single d20 is hugely risky when you have only 1 shot to make the roll. One can't apply the same standard to determine DC for skill checks as you do for combat DCs.

I can't say for sure how much thought the 3E designers put into recognizing the difference in curves between the two systems. In fact, I'd say very little since one of the huge selling points of the game was a "unified system" for everything, and most players would quickly point out that the "system" for handling skill checks is identical to the "system" for handling combat.

Fact is, however, its not. Its very very different in a way that aspiring game designers should spend a lot of time being aware of in their own designs. There may be a very good game enhancing use for using two seperate probability curves in your design. But its something you should do on purpose, not by accident.

THat's true, as long as you see a whole combat as one "event" to be rolled. Many gamers don't see it that way.

Consider, for example, a climb. In DnD, this is not handled by one skill roll. This is handled by multiple skill rolls (one per 30' of climb, roughly) in which any one failure means the failure of the whole endeavor.

There are other examples of multiple skill rolls being required in 3.0 and 3.5 DnD. Crafting, for example.

Many gamers don't see it that way.

Consider, for example, a climb. In DnD, this is not handled by one skill roll. This is handled by multiple skill rolls (one per 30' of climb, roughly) in which any one failure means the failure of the whole endeavor.

The penalty for failing any one climb roll is not necessarily falling. Unless you fail by five or more, you just fail to make progress.

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?

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.

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.

Vaxalon wrote: 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.

Noon wrote: 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.

Kesher wrote: 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?

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.

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.

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

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

Kesher wrote:
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...

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

Callan wrote: 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.

He further wrote: 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.

Ralph wrote: 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.

Kesher wrote: 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?

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.

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.

mindwanders wrote: 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?

M. J. Young wrote: 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.

Correct me if I'm wrong, but doesn't WW have round-by-round combat, in the same manner as DnD? It has been a long time since I've played, but I would recall if something more radical had been included.

MJ, yes. They are certainly not identical. The d20 die pool has an interesting effect that you aren't sure exactly how many dice are going to be in the pool at the end (i.e. how long the combat is going to last), and the ability to signicantly change the nature of the conflict in the middle is also different. But the effect of a bell distribution of expected results over time is the area I was drawing attention to.


Callan, I tend to think of the Take 10 rules as something of a patch. Its a jerry rig solution to the issue that works and is simple. The purpose is to allow the player to simply choose to get the expected value as opposed to actually coming up with a system that reliably produces the expected value. It would be interesting to know whether the designers actually thought about this rule in terms of probability and distribution curves, or if it was just a more instinctive reaction to too much whiffing in the skill rolls.

If you look at the designer's notes, the take 10 and take 20 rules were created in order to reduce handling time.

My approach in Chaos & Order is to make the amount of fortune a 'game tool', a variable that can be changed in the game.

The basic idea is to always throw 2 dice, the plus die and the minus die, to this is added the trait value of character and then compared to difficulty rating. The thing is, the die to be used can be d2 or d20, or anything in between. The requirement is the fact that player needs to clarify why the particular die is used, gm can veto.

I just put my game, Skein, up for you theorists to review, which has what I think is a fairly unique dice handling system in it.

Each competency has a rating, an even number between 4 and 10, that describes the "size" of the dice that are used for it.

The "chaos level" of the situation determines how many dice are rolled. High chaos equals low number of dice. Combat, for example, is usually a high chaos situation, where luck plays an important role.

Let's say you've got an ogre, very dangerous, whose "bashing things over the head" competency is 10, and a halfling, whose "stabbing things in the kneecap" competency is 6. When they go head to head in a wild, riproaring fight, the ogre rolls 1d10 and the halfling rolls 1d6.

On the other hand, when the halfling, with his competency of 10 in "get into places people want kept secret" tries to pick the competency-8 lock, he rolls 3d10 against the lock's 3d8.

Does it allow you to take rigerous planning on the part of the players into account (cyberpunk style casing of the joint before sneaking in and attacking the ogre while he's doing something else)?

For some reason, one of the bigger blind spots in experience seems to be exactly this:

- It's very unlikely that a player will roll X. This means that it's very unlikely that any player in the group will roll X.

Of course, this is all wrong. Some common things that occur due to this are:

- The characters are all sneaking past a sleeping guard, which is meant to be very easy. Just for fun's sake, the GM asks everyone to throw a d10, with "1" meaning their character will somehow stumble. Seems innocent enough, except that in a group with five characters there's actually a 41% chance of at least one character somehow stumbles. This is a common reason for the "the characters always screw up at crucial points" phenomenon.

- There's something that's "extremely hard to spot"; it would need a roll of 20 on a d20.... Now if there's for example six characters who get to try this, the chance of them noticing this "extremely hard to spot" thingie would be 1 in 4 and then some. That's usually a lot more than what the GM had in mind.

The "power in numbers" thing get truly ridiculous in systems with large dice pools like WW:s Storyteller. But then again the Storyteller system sucks like no system I've used. Goes to prove that having good mechanics which fit the game theme is by no means necessary to sell a game.

mindwanders wrote: Does it allow you to take rigerous planning on the part of the players into account (cyberpunk style casing of the joint before sneaking in and attacking the ogre while he's doing something else)?

Itse, good call.

Back in the day when I was GMing campaigns that made regular use of spotting checks I just had the character with the best skill make the roll. If the most perceptive character misses it, its unlikely that the less perceptive character would notice it (assuming equal attention being paid). This also provides some additional niche protection.

Similiarly when sneaking I had the most clumsy character roll on the theory that if they succeeded, everyone less clumsy would have also.

LordSmerf wrote: Here are 3 major types of probability curves...flat curves...bell curves...[and] modified bell curves when rolling multiple, differently sized dice

Itse wrote: - It's very unlikely that a player will roll X. This means that it's very unlikely that any player in the group will roll X.

Of course, this is all wrong. ...

The problem with that, Ralph, is that then what's the use of having that perception score for my character, if yours is one higher? For when we're not together?

Hero Quest handles this well. The "lead" player uses his total score, and the other players augment, meaning that they add one tenth of their ability. This keeps the niche protection, and allows the other scores to be useful. And when they split up, like I suggest above, then each individual score can be counted on.

The fact that this is encoded as the normal method in the rules is a godsend.

Mindwanders, this is precisely the dangerous thing that one should not say about these curves. First, yes, the "expected value" is 150. Meaning that if you make the 100d2 roll many times, you should expect to get 150 per roll or so. And it is the most common result as well. And, yes, the tails get extremely small. But the fact of the matter is that the standard deviation on this is not so small as you might think. First off, the chance to get 150 itself is actually very small, less than 8% (as opposed to about 1% for a flat curve covering the same range).

But more importantly, though most of the rolls will range from 145 to 155, every third one will be outside that range. Given that the results of this roll are 100-200, that's a ten percent range for just the common results. Whenyou use the oft quoted 95% of results, you get around from 140 to 160. Yes, that means that, during play you're likely to see scores ranging from 140 to 160 every night. Moreover, there's a chance that you might see from 136 to 164 in a night. And you'll likely see something a little outside that range once every campaign. Yeah, you'll probably never see anything lower than 130, or higher than 170 (only one in 25,000 rolls or so), but the result is hardly what I'd call "karmic."

Mike

Mike Holmes wrote: Hero Quest handles this well. The "lead" player uses his total score, and the other players augment, meaning that they add one tenth of their ability. This keeps the niche protection, and allows the other scores to be useful. And when they split up, like I suggest above, then each individual score can be counted on.

Mike Holmes wrote: ...you'll probably never see anything lower than 130, or higher than 170 (only one in 25,000 rolls or so), but the result is hardly what I'd call "karmic."

I'd like to point out the obvious:

The more sides on a die, the more random the outcome. Thus, using a D12 is more random than 2d6, which is more random than 3D4, but not only because there are more dice. A d4 can only produce 4 results, a d6 can produce 6, etc.

Also, the more that is added to a roll, the less random it becomes, in terms of overall variation. Thus, rolling a d4 generates a result between 1 and 4. Rolling d4+100 generates results of 101 to 104.

So, in summary the dice rolling convention used by D20 is more random than that used by GURPS for instance. Rolemaster is the one of the most random (it's two dice count as one) and D6 has a sliding scale. Not that these are particularily good examples, but this is what is most important about dice mechanics: that they are inline with your design intent.

Just wanted to make sure that the obvious remained so.

Cheers
Jonathan

ErrathofKosh wrote:
Also, the more that is added to a roll, the less random it becomes, in terms of overall variation. Thus, rolling a d4 generates a result between 1 and 4. Rolling d4+100 generates results of 101 to 104.

Vaxalon wrote:

ErrathofKosh wrote:
Also, the more that is added to a roll, the less random it becomes, in terms of overall variation. Thus, rolling a d4 generates a result between 1 and 4. Rolling d4+100 generates results of 101 to 104.

This may seem obvious to you, but it's not true. The standard deviation, which measures the variability of an event, is not affected by flat modifiers to a die roll.

Sorry, I must be more clear on what I meant. Perhaps I should say "the range of the result" versus it's "scale". The range remains the same while the scale becomes larger. So, I care if I get a 1 versus a 4; but, usually it doesn't matter that I got 64 instead of 61. Thus, while you're right that it doesn't change the relative size of the standard deviation, it does move it, which can be important.

In a lot of RPG's, a higher result is better. If my goal is to roll over twenty, I will do so more often if I add ten to my d20 roll than if I add five. In fact, if I add 20, I will always reach my goal and the roll becomes meaningless, thus less variation in my success/failure result, not in my numerical result.

Sorry if I was unclear, as I sometimes hit the submit button a little too quickly. (I'm still working on that...) :)

Cheers
Jonathan

EDIT: x-posted with John Kim

Yes, in many situations, the die roll can become meaningless if the modifier is high enough. When that happens depends strongly on the exact implementation, the target number and the bonus together.

Valamir wrote: Back in the day when I was GMing campaigns that made regular use of spotting checks I just had the character with the best skill make the roll. If the most perceptive character misses it, its unlikely that the less perceptive character would notice it (assuming equal attention being paid). This also provides some additional niche protection.

Similiarly when sneaking I had the most clumsy character roll on the theory that if they succeeded, everyone less clumsy would have also.

you wrote: The more sides on a die, the more random the outcome. Thus, using a D12 is more random than 2d6, which is more random than 3D4, but not only because there are more dice. A d4 can only produce 4 results, a d6 can produce 6, etc.

M. J. Young wrote: Also, the die rolling convention of d20 in the D20/3E version of D&D is only more random than GURPS at low levels. As characters progress, the increased bonuses reduce the importance of the random factor. Whether d20+100 is less random than d20+0 depends entirely on the range of possible target numbers. If both are against the same range, then at least one of them is not random at all--either you can't succeed or you can't fail, which is clearly not random. But if you assume that the difficulty might be anything from 1 to 200, then there are a lot more situations in which the d20+100 will automatically succeed, which means randomness is out of the system when the target number is not within 19 points of the bonus.

2d6 is less random (more predictable, lower standard deviation) than 1d12, for all the reasons already discussed.

However, the result of 3d6 summed is more random (less predictable, higher standard deviation) than 2d6 summed which is more random than 1d6 summed, despite the first having a bell-shaped curve and the second having a pyramid-shaped curve. Adding more dice to a summed die roll always makes the outcome less predictable, because the increase in the possible outcome range always more than compensates for the bell curve trend favoring "average" results. As you add more dice, the peak of the bell curve becomes broader and lower, not sharper or higher. The standard deviation increases.

With d26 I can predict a result of 7 and be right 0.167 of the time. With 3d6 the best prediction I can make will only be right 0.126 of the time. Allow me to predict within a range of four numbers, and I can be right 0.482 of the time with 3d6 (choosing 9-10-11-12) but I can be right .556 of the time with 2d6 (choosing e.g. 5-6-7-8) and .667 of the time with 1d6 (choosing any four numbers). This rule that adding dice decreases predictability and increases the standard deviation holds true for any range of prediction, for any number of any size (or combination of sizes) of summed dice.

What makes people believe that the results become more predictable ("less random") with more dice is that if the overall range of outcomes is held the same or nearly the same, then more dice (summed) are indeed more predictable. (The same is true if the roll totals are somehow being normalized relative to the range of outcomes, but that is rare in role playing systems.) A roll of 10d10 summed is more predictable than a roll of 1d100, and a roll of 10d100 divided by 10 is more predictable than a roll of 1d100. But that doesn't mean that adding more dice to a given roll makes the results more predictable, as is often claimed. If someone describes a system in which 1d12 is rolled, complaining that it's "too random" and asking what to do about it, one person is likely to recommend replacing the 1d12 with 2d6, and another is likely to recommend replacing the 1d12 with 2d12. The first advice is right, the second is dead wrong.

The other pitfall of judging relative randomness or predictability is that one must look at the actual results as applied in play, not just at how the dice might fall. In a roll for number of hit points of damage done, 3d6 is more predictable than 1d20 and way more predictable than 1d1000. But if the roll is for a simple success or failure, it's meaningless to say that 3d6 (succeed on 11+) is "less random" than 1d20 (succeed on 11+) or 1d1000 (succeed on 501+). Regardless of the differences in the distribution of numbers rolled, they all have the same distribution of outcome; to wit: 0.5 success, 0.5 failure. The same is true if the rolls are opposed rolls against another roll of the same type, higher total wins (except that the chance of a tie will be a bit different). If the rule is that ties are rerolled, then my chances of winning with my 1d4 against your 1d4 are the same 50-50 as if we're each rolling d100 three times, multiplying them together, and raising to the power of the phase of the moon. Neither is in any meaningful way "more random" than the other.

I've ranted before about the general lack of understanding of the behavior of modifiers in game design. The effect of a modifier on a simple success roll can be understood in terms of its additive change to the probability of success, the factor by which it multiplies the probability of success, or the factor by which it multiplies the probability of failure. Of the three, the first is the least justifiable in terms of representing in-game-world causal factors but it's by far the most often used to base modifiers on. That's why cumulative modifiers tend to lead to problems, and why at higher levels it's hard to find a monster that the magic-user has a chance to hit that the fighter (of the same level) has a chance to miss.

- Walt

There is no such thing as more random a thing is random or it is not. A 1 in 4 chance is no more random than a 1 in 2. Having three outcomes doesn't make it more random than having two.

Actually, Jack, you're right... a 1d4 and a 1d2 are very, very similar when it comes to "randomness".

However, wouldn't you say that 1d10,000 is more "random" than 1,000d10? In the first case, you can't say anything about the outcome other than that it will be between 1 and 10,000, whereas in the second case, you can be pretty certain that the result will be between 5,000 and 6,000.

We're not so much talking about the SIZE of the dice, as we are about the NUMBER of them. The more dice, the more likely the results are to cluster around the average, and thus the more predictable the outcome.

Vaxalon wrote: However, wouldn't you say that 1d10,000 is more "random" than 1,000d10?

Hello,

It's time to break out sub-topics of this thread into threads of their own. I'm starting to see little island-conversations instead of a discussion, and a few too many one-line replies to single points. That's a good sign overall that the topic is catching a lot of different interests, but it's also time to start spawning.

H'm - "spawning" - I use that term a lot, ought to explain it. I use it to mean the people involved in the thread taking the responsibility to bring a sub-topic to a new thread, properly referencing the original, and without returning to post on the old one, letting it lapse. As opposed to "splitting," which is what I have to do with the thread when people don't spawn.

Spawning = sexual reproduction, splitting = asexual reproduction, and now I know I better stop explaining this point. Spawn now, please.

Best,
Ron