Dice, Probability, and Simulation

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    Hello Friends!

    I’ve been working up to restarting my old thread once again, with some new material to boot!

    Before cracking into that new material, perhaps a primer is in good order? Here’s a really great blog post on using the 3d6 normal distribution or “bell curve” dice:

    3d6 is not less swingy than d20

    …I’ve not taken the time to let all of the article sink in, but it does remind me of the many d6 probability charts and graphs I prepared for the old thread. Just like in the case of the old thread, you need to be careful when using the information. It can take a good bit of work to understand it all, but if you want to come up with something new, shiny, and awesome, a bit of research and modeling will be needed in order to make something which not only works, but “gels” well with the intellect. I get the impression that the majority of games and systems work well enough, but they often fail to gel. Ergo, we have Codex so that D&D gels, and not just works.


    Hans Hellinger

    I haven’t read the whole article, but I don’t buy the premise. We did the “monte carlo” or “Casino” method, i.e. brute force, when testing Codex. I.e. we rolled the various combinations of dice 10,000 times.

    3d6 produces a vast amount of 8-12 results compared to 1 or 18. This was the basis of the ‘modified dice pool’ or ‘roll many / keep one’ method we used in Codex, partly because people get so angry about the flat curve of a 20 sided dice, especially when they roll a ‘fumble’ result.

    Multiple 20s is ideal for me because it gives you the wide range of potential results (1-20) with the bell curve outcome you want (skewing higher for more dice, with much less possibility of a ‘natural 1’. 3 x D20s meant you got a ‘natural one’ something like 1 time out of 3,000 die rolls if I remember correctly.

    I could set up the simulation again to test it (again) easily enough.


    Alright, to be perfectly honest, it looked like a really interesting article, so I posted it here to take a better look at it later! It’s going to take some time to do that.

    First and foremost, you need to be aware of the fact that the article is full of “math poop,” which is a technical term. I take a long time combing through my own original material because making sense of the shite is pretty challenging. I’m long past calling myself any semblance of intelligent, but it still hurts to be called retarded – unless the person calling names is me. :p

    …So, when you see “variance,” it is not in the common English word context. It’s in the math context, which can be just as bad as evil-robot-lawyer-speak if you don’t know what’s going on. I actually still don’t know what’s going on, but I intend to get it sorted out by no later than this weekend.

    One thing you can start with is the first set of graphs the blogger puts out. The middle one is indeed the normal distribution output of 3d6, which is a cumulative dice. Because this is the most readily read graph (it makes sense when you look at it), we can dissect what all the axes are on the graph, and therefore see what they are on the other graphs as well:

    1. The bottom, horizontal axis is the die output, or in this case, sum. 3d6 ranges in values from 3 to 18 for a total of 16 outputs. Likewise, the D100 (kind of but not exactly a cumulative dice) graph to the left ranges from 1 to 100, and the non-cumulative d6 chart to the right is a bit different: it counts the number of d6’s used for a given roll.

    2. The vertical axis notes the probability of getting a given roll. 3d6 definitely spikes in the middle, because on average your rolls will sum to those numbers, regardless of how the sum is achieved (I suppose you could design a game in which the manner of getting the sum would be important, but why (or maybe, why not)?). Ideally, you have a 1% chance of rolling any one number with a D100, but I don’t buy it when you’re using crappy dice like D10s. However, note that the D100 graph is a constant flat line, which is what any other single dice would be as well in ideal circumstances. The non-cumulative d6 chart for Shadowrun on the right is again different, as it’s noting the probability of scoring with a 5 or 6 with any one die. The curve for that chart drops off not because you’re less likely to succeed with more dice, but because you’re less likely to succeed with all of the dice at once. The Shadowrun graph is going to be what is most similar to Codex in this case.

    3. All of the graphs have an added pair of numbers on them, which I believe are the variance and standard deviation. These are important numbers apparently, but I still don’t know what significant they hold for real human beings. I will get back to you on that matter in time…

    Hans Hellinger

    Hahaha ok thanks for the translation.

    I saw the bell curve on the 3d6 dice, which is what I expected, and the Shadowrun dice pool, so I agree the provocative title must have more meaning in ‘math speak’ than in human speak, so to speak 😉


    Right, so a bit behind on following up on things. However, I do want to clarify the Shadowrun / dice pool chart presented in the linked articles. I looked over my old tabulations from the outdated forums – check out the p = 1/3 chart (meaning that 5 and 6 on a d6 will work; these are two of the six faces, hence 1/3). You will note that if you compare the die rolls column (n = 8) with the success rows (k), you get the same data as the chap in the linked article got:


    Compare this with the other graph:

    k = x = 0, 3.90%
    k = x = 1, 15.61%
    k = x = 2, 27.31%
    k = x = 3, 27.31%
    k = x = 4, 17.07%
    k = x = 5, 6.83%
    k = x = 6, 1.71%
    k = x = 7, 0.24%
    k = x = 8, 0.02%

    …Note that it’s really important to understand data in context. The graph he generated for Shadowrun’s dice pool requires that all eight dice be rolled to get that data. If you require a certain percentage (p = ?) to meet a target number with a given number of dice (n = ?), you need to have numbers available for those situations, and you need to understand how to use that information once you have it.

    One thing to consider with Codex is that the target number can change depending on the scenario, rather than being a fixed percentage. The dice can also change in terms of quantity used. So, you will have similar charts generated to the one for Shadowrun in that particular case, but they will all be very different.

    Hans Hellinger

    I’m going to have to read that another 3 or 4 times to see if I can understand it.



    I was able to steal a few minutes to work on this – is the following chart clear to you?


    In the future, I will need to sink more time into visuals! I took the relevant parts of my table and the other dude’s (TM) graph and made some more-angular-than-squiggly lines to connect them to each other. Look at that, then read my prior post, and see if it makes sense to you now. If it’s still unclear, I will try something else.

    Hans Hellinger

    No I don’t understand it, can you explain a bit more?


    From my reading of it what I think the premise is has nothing to do with bell curves and flat line probabilities and everything to do with outcomes in RPGs. I believe the primary conclusion they are making is that flat or curved probabilities makes no difference when the target numbers are appropriately set. In other words if you want a 50% change of success and set a TN around 11 with 3d6 or set it to 1-50 on a single d100 roll then you’ll get what you are looking for. I think the suggestion is that the whole bell-curve -v- flat probability ends up just being a red herring on how a roll comes up because if the final outcome ends up being the same based on how you set the target then it really doesn’t matter which method you use. Sure, if folks FEEL they need the bell curve for some reason, the that’s cool, go with that, but in the end it’s probably just a wash compared to what you are effectively looking for in the final outcome so long as that outcome is aimed to the roll type, so in that sense the “swinginess” ends up simply being irrelevant. Yes, 3d6 are less “swingy” than a straight percentile in the roll, but the roll is not the relevant part, it’s the outcome you are concerned with, after all.

    That’s what it looked they were saying to me, anyway. (And I am definitely no math major, or anything).

    Hans Hellinger

    Ah, I thought that is what they are saying, and thanks for explaining it… but I don’t agree! I think you get much more ‘middle’ numbers with an ordinary bell curve (i.e., 3d6 vs. 1d18) and you’ll skew much higher AND almost never roll a ‘natural 1’ if you pick the highest out of multiple dice. Maybe it’s time to run another ‘Monte Carlos’ simulation on my computer to make the point.

    Hans Hellinger

    “Yes, 3d6 are less “swingy” than a straight percentile in the roll, but the roll is not the relevant part, it’s the outcome you are concerned with, after all.”

    This is the part I don’t get, I don’t understand this distinction. I’m writing the program to do the ‘Casino’ test now.


    Well, I don’t think that article is trying to cover the end cases at all (how often you critically succeed or fail). That’s a different argument than I think they even bother trying to touch on. I think all they are trying to cover is that if you set the right difficulty/target number for something then the statistical chance of achieving that outcome (the binary one of strictly success or failure, not one of where on the middle or the edges the numbers fall) would be the same. Well, ok, good to know as a game designer that you need to be aware of designing proper targets for outcomes, so that says to me you are then free to look at the type of rolls you want to use for other uses. If you ALSO want to be able to affect things outside of success/failure then using a pool like Codex does instead of a straight d20 definitely affects that, and uses that to the advantage of the system. Or more accurately, it allows the player the option to make that decision (since they can always choose to roll with just 1 d20 or multiples, after all).

    Hans Hellinger

    Ok … after thinking that over a bit more, I think I DO get what that guy was trying to say, but I don’t think it’s relevant. I’ll explain why in a minute.

    Here are the results of my first test. For this first run I’m only doing 2000 rolls at a time because my program is pretty crude and it runs slow, I’ll fine tune it a bit later. Messing around with this already forced me to address a nagging rounding error on my random number generator for the character lifepath application, so that’s a positive side benefit even though it took me 3 hours to fix it + write this number cruncher thing.

    So ok anyway, this is a ‘Monte Carlos’ test of Roll many / Keep one, keeping the highest number of dice rolled out of one, two, three, and four. I’m tracking the number of Critical Failure and Critical Success results which are key for reasons I’ll explain in a followup post.

    Anyway, here are preliminary results.

    Roll ONE die

    Number of Natural 20: 76 (3.8%)
    Number of Natural 1: 83 (4.1%)
    Iterations: 2000
    Cumulative total: 20786
    Average: 10.39

    Roll TWO dice

    Number of Natural 20: 125 (6.2%)
    Number of Natural 1: 52 (2.6%)
    Iterations: 2000
    Cumulative total: 23333
    Average: 11.67

    Roll THREE dice

    Number of Natural 20: 206 (10.3%)
    Number of Natural 1: 18 (0.9%)
    Iterations: 2000
    Cumulative total: 26539
    Average: 13.27

    Roll FOUR dice

    Number of Natural 20: 241 (12.05%)
    Number of Natural 1: 5 (0.25%)
    Iterations: 2000
    Cumulative total: 28063
    Average: 14.03

    Hans Hellinger

    Hey sorry Zarlor I was working on my reply before your post showed up.

    So here is how I see it. What his saying is that 3d6 will have the same average as 1d18. Which is technically true. In both cases your average is going to be about 9 or 10 or whatever. But what matters most for players from my experience is not being flogged with the flat curve. They want to avoid the negative outiliers. So with 3d6, they want to avoid rolling a 3. And 3d6 would be better for this than say, determining a number between 3-18 via percentile dice or something.

    I’ll post some hard numbers on this later when I have time to do a bit more coding, (and I’m sure you know this already but bear with me as I restate the obvious), but basically a bell curve result such as you get from 3d6 means that most of the die roll results you are going to get are going to be in the middle range, 8-13. If you add up the number of results you get from rolling 3d6 10,000 times, something like 2/3 will be in that middle range, while the higher and lower results will be about 1/3. The chart above matches that ratio I think.

    This gives you the experience of having far fewer outlier results. The curve skews toward the middle.

    With the ‘advantage’ system as you call it (though I am loath to do so!) the curve skews more and more toward the top, and increasingly diminishes the chances of a ‘critical failure’ That is what matters most in my experience especially for your more tightly wound or angry players (like one or two people in your group): NOT getting a critical fail. The baseline with a 20 sided die is about 20% chance (around 4% in my test but that is because I used a small sample, it evens out the more iterations you do).

    Rolling two dice reduces that critical fail chance already to about 2.5%, which helps (that is half of the normal risk), but 3 dice reduces it to 1%, a big improvement (a fifth of the normal risk). At four dice the chances are one quarter of one percent. This allows those players who can’t stand to get a Crit Fail to really avoid them if they are willing to spend the dice (or manage to min-max the situation to arrange for enough extras). The average die roll also goes up, but kind of gradually, as does the chance of a crit success, but still not to the point that it becomes routine (it becomes around a 1 in 8 chance based on my last test, though I think it ends up being about 1 in 6 or 7).

    So this is what solves my problem. Skewing the numbers higher, as an option which players ‘pay’ for in one way or another. Some people like to gamble, and don’t mind the flat curve. Others can never get used to it, they feel like something is wrong and I’d say there is. Your chance of dropping a sword really isn’t one in twenty if you are paying attention to what you are doing. Especially if you are experienced. That’s why it feels off.

    Hans Hellinger

    So bottom line, 3d6 is “less swingy” than one d20, though I guess it depends what you mean by that term.

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