A Mathematician Speaks: Rolling dice

[Vaxalon]

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.

[John Kim]

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.

Correct mathematically.  However, for game purposes the absolute standard deviation isn't necessarily important.  What matters is how those numbers are translated into game effects.  One issue is comparing the variation of the modifiers to the standard deviation of the die roll.  i.e. If armor classes go from 10 to 150, then a 1d20 roll isn't much on that scale.  However, if armor classes go from 10 to 15, then the 1d20 is the dominant effect.  There is also the question of how the numbers are translated into effect.

[ErrathofKosh]

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

[Vaxalon]

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.

[M. J. Young]

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.

Those concepts are expressed as rules in Multiverser.

Jonathan, to try to get to your first point,

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.

I see what you're saying, but there's faulty thinking there.

Yes, a d4 is less "random" than a d12, because it produces fewer permutations. But 3d4 is less "random" than d12 because of the process of creating the curve.

First, if you roll 3d4 you can't get lower than 3; so the range has been reduced from 1-12 (12 steps) for the d12, 2-12 (11 steps) for 2d6, to 3-12 (10 steps) for 3d4.

Yet that 3d4 3-12 range is less random than the same 3-12 range created by d10+2.

3d4 has (4x4x4)64 possible permutations; d10 has 10.

Of those permutations, only one produces 3, and only one produces 12. That means your chance of rolling the extreme is 1/64 (for each of high and low), about one and a half percent. Your chance of rolling those same values on a d10+3 is ten percent.

The center point is 7.5; you can't roll 7.5 with either method, so we'll use 7 (which has the same probability of occurence as 8). With d10+2, you roll a 7 exactly one time in ten, ten percent. With 3d4, I thumbnail twelve permutations that get you a total of 7--which means that roughly 35% of your rolls will be 7, and that means that 70% of all rolls will be either 7 or 8.

Since the remaining 30% will split evenly above and below, that means that 85% of all rolls on 3d4 will be at least 7; on d10+2 40% of rolls are lower than 7, so 60% are at least 7. Although it's not really intuitive, you are more than 40% more likely to meet or beat 7 with 3d4 than you are with d10+2 (because 85 is more than 40% greater than 60). So the reduced randomness is not related to range so much as to permutations.

To offer another way of looking at it, let's roll 3d10. There are two ways 3d10 can be read easily enough. One is to sum the dice, a curve of 3 to 30. The other is to use marked dice, one times a hundred and one times ten, to produce a d1000 (technically, 000 to 999, but traditionally we would treat 000 as 1000). Either way there are a thousand permutations--the dice can come up any one of one thousand different ways. However, in the d1000 arrangement, each of those permutations is an individual value with its own equal chance of success--whether it's 1 or 75 or 369 or 987 or 1000, it has one chance in a thousand of being rolled. By contrast, the 3d10 additive approach means that seventy-five of those permutations are equal to 16 and another seventy-five are equal to 17--one hundred fifty out of the thousand possible rolls, or 15%, land on one of those two center numbers.

I once had a guy try to replace the 3d10 GE roll in Multiverser with X number of consecutive coin tosses. I worked out that you'd need ten such coin tosses, and you'd have to write them all down and use a table to interpret them, to replace the dice. Curves can be very tricky.

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.

--M. J. Young

[GB Steve]

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.

With 3e though, the target numbers tend to scale up as the bonus does so the randomness tends not to be reduced.

The main thing that happens as you go up levels is that you extend the range at the bottom of things at which you are bound to succeed. But then you find that these things, such as easy locks or things to kill, either tend not to occur much, or you ignore them (which is functionally equivalent). So whilst the +20 gives the appearance of power, there's an equivalent -20 from the bad guy that keeps you where you were.

The main difference between levels in 3e is your choice of feats which offer much more of an advantage than most other gains and do more to differentiate characters (this is also the source of my main gripe - you can't do the range of things that the genre demands - but that's for another thread).

[Walt Freitag]

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

[Jack Aidley]

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.

[Vaxalon]

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.

[Jack Aidley]

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

Nope.

I might say it is less predictable, has a higher standard deviation, or has a higher variability. But without a context in which the roll is being made none of these judgements have any relevance at all, and even with a context the term 'more random' has no well defined meaning - randomness is a binary either-it-is-or-it-isn't affair.

Although to be honest, I might use the term 'more random' I would understand that it is sloppy use of language and not something that has any place in a serious discussion of dice mechanics. As such I think we'd be much better served discussing things in terms of probabilities.

[Ron Edwards]

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