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Thoughts on Rolling Dice (Uniform vs. Normal Distribution)

To the singularity, and beyond!

My main mode of playing a TTRPG at the moment is running a 5th edition Dungeons and Dragons campaign online for a group of friends. Prior to this (and skipping past the hiatus I'd had from playing), it was mostly playing 2nd edition Advanced Dungeons and Dragons. The two systems have myriad ways in which they're similar, specifically with regard to approaching resolution mechanics outside of simple attack rolls and saving throws (which are abstract enough in what they're modeling that a simple uniform probability distribution is generally acceptable and for which the outcome of any single instance as a success or a failure is rarely a cause of cognitive dissonance). While it's partially a testament to how well the old mechanics have stood the test of time, there's a case to be made for integrating character competence into the resolution mechanics by using a die roll with a mode rather than just using a d20 roll. Let's start unpacking what I said there and what it means for me.

This post is going to get into some mathematical concepts out of necessity. I'll try to explain it in a way that doesn't require past education beyond simple arithmetic and practical knowledge, but as I am someone who understood how "2 + 1 = 10" could be correct from reading that I did for fun as a child, I make no promises that I can achieve that. However, if you're reading this blog post, I trust that you're capable enough with using the Internet that you can find some introductory lessons on probability on your own.

For simplicity's sake (and to keep things in line with most of my experience), let's put aside dice pool systems that compare a number of successes vs. failures in favor of focusing on either a single roll that needs to beat a target number or opposed rolls where whoever gets the highest/lowest result is successful. There are reasons why those are my personal preference beyond just familiarity, but this introduction is taking long enough already without those details.

When it comes to rolling dice in TTRPGs, it's useful to have some idea of what results can be expected. As a player, combining that with whatever target number you need to beat for success that lets you gauge how likely you are to succeed at something (or how many tries you'll probably need). As a game master, knowing what results your players will likely get when they are attempting whatever actions they're trying lets you pick reasonable target numbers. All in all, it's very useful information.

Unfortunately, the human brain is quite horrible at processing statistics intuitively. Certainly, after years of experience with it, I can say readily enough that beating 6 on a 2d6 roll, beating 10 on a 3d6 rolls, or beating 10 on a 1d20 roll are all roughly equivalent (the actual odds are 58%, 50%, and 50%, respectively, shown below in black, yellow, and blue, respectively, based on this link).

I've got nothing cute to say about probability graphs

It should be obvious that any individual face of a d20 has a 1-in-20 (or 5%) chance of coming up, assuming a fair die. This is what's known as a uniform probability distribution. Thus, given that there are ten results which are higher than 10 (11, 12, 13, 14, 15, 16, 17, 18, 19, and 20), the odds of beating a 10 on a 1d20 roll are 10 x 5% = 50%. However, as anyone who's played a d20-based TTRPG for an appreciable length of time can attest to, the fact that each result has the same chances of occurring doesn't mean that you won't see streaks of three or four (or more) rolls that all get the same result. This is because the probability is what we can expect to happen based on the results of hundreds if not thousands of rolls, but it's purely theoretical because each individual roll is unique. In fact, in modeling a set of 2000 results of a 1d20 roll in Excel (which I did to check the impact of the advantage/disadvantage system in 5th edition D&D), no matter how many times I mashed on F9 to repeat the randomization, even with rounding off the percentage to a whole number, I never saw each result actually come up 5% of the time. Most of them did, but there were always a few which actually came up 4% or 6% of the time, despite the large sample size.

Compare to rolling 3d6. Each individual die has a 1-in-6 (or 16.7%) chance to give any individual result. However, because we're taking the sum of three separate dice to get the roll's result, there's a tendency for the results to cluster around a certain "most common" value (know as the "mode") since a high result on one die will tend to be counteracted with a low result on another die. While there are 216 total results for rolling 3d6 if each die is counted individually (each with a 1-in-216 chance of occurring), there is only a 1-in-216 (0.5%) chance of getting a total of 3 or 18, only a 3-in-216 (1.4%) chance of getting a total of 4 or 17, and so forth. This is what's known as a normal (or bell-curve) probability distribution. The results are still random, but there is a higher likelihood of getting something in the middle of the total range than something near the limits.

When it comes to situations that are very abstracted and where we expect luck to be a significant factor, such as trying to land a hit in combat (attack roll) or trying to resist a magical effect (saving throw), high randomness is fine. This represents a high degree of uncertainty, which seems natural in the chaos of a fight or when tangling with forces from beyond rational understanding. It can also seem natural in cases like what came up during the session that I ran last night, where the characters were trying to climb into a cave entrance on a cliff-like rocky hill face. Descending into the opening was not very difficult, but since none of the characters had a background with notable experience in rock climbing, they still needed to make a DC 10 Strength(Athletics) check for someone to get in and set up a rope that the others could climb. Despite Maya (the team's fighter) having a +3 modifier to that (+1 from her Strength score and +2 from proficiency in Athletics), thus having a 70% passing it (due to fourteen die results providing the necessary 7 or higher), she still failed twice before making it in, causing her to take some damage from falling. It was unlikely, but it was still sensible.

One case where this can break down into cognitive dissonance is when it comes to opposed rolls. Suppose Adalynn (the team's waif-like sorcerer, who has a Strength modifier of -3) was mind-controlled into trying to restrain Maya physically. In the rules for 5th edition D&D, this requires Adalynn to make a Strength(Athletics) check (for which she has a -3 modifier) that beats Maya's result on a Strength(Athletics) or Dexterity(Acrobatics) check (for which her best modifier is +4). On paper, it sounds like it should be a waste of time. Adalynn needs to beat Maya's result by 8, which means there's a 40% chance that Maya's roll will be high enough to win automatically (13 or higher), and there's a 35% chance that Adalynn's roll would be low enough to fail automatically (7 or less), and even in the spread of results between those extremes, Maya would have a significant advantage. However, because the d20 results follow a uniform distribution, there remains a 19.5% chance that Adalynn succeeds (link for evidence).

Another place where this can break down is when it comes to a character's assumed competence compared to their actual roll results. I played through the free solo Call of Cthulhu adventure Alone Against the Flames a couple of times recently, as a way of getting introduced to that game system (and I will have a review of it posted soon). Call of Cthulhu runs on a d100 system, thus any die result has a 1-in-100 (1%) chance of happening. My first attempt at the adventure was with a scrawny professor (50 Strength, 50 Constitution, 40 Size). My second attempt was with a brawny private investigator (70 Strength, 60 Constitution, 50 Size). Both ended up in a situation where they had to roll their Strength or less in order to break out of some restraints to save themselves from being burned alive. The professor managed to succeed, even with a penalty die (similar to disadvantage) which meant a 25% chance to succeed. The private investigator failed, despite a 70% chance to succeed. It was fun, but I can't deny that I had a little bit of frustration with the second case, particularly because I had passed a check a moment earlier to have an NPC tell me that the restraints had been weakened (which added to my frustration because I had the previous experience to feel like it was a case of illusionism, but that's a topic for another day).

Certainly, I recognize that randomness leading to unexpected outcomes is a good part of TTRPGs, since that's a component of what makes them different from just writing stories. However, there are cases where it can be too random, overpowering the player's choices in an unsatisfying way. Again, I'm fine with this to some degree, and when I'm running a game with a situation where a highly competent character is faced with a simple task without any external pressures, I have no problems with giving them an automatic success instead of requiring a roll, thus minimizing exposure to the most extreme problem cases. However, I do think that the uniform distribution is flawed when it comes to dealing with ability/skill/etc. checks, because I prefer to assume the characters are competent at easy-to-average tasks. Thus, in the 2nd edition retroclone that I've been working on, I replaced using 1d20 for ability checks with using 2d10. This retains almost the same range of results, but it has an actual mode (11) and a normal distribution, which means that "average" performance is supported in the resolution mechanic and the extreme results (2 and 20) only come up 2% of the time rather than as often as any others.

Incidentally, I came across this website from 2001 yesterday which made similar remarks and came to the same final conclusion with respect to d20 rolls compared to other systems (3d6 or 2d10). It's worth a read if you want another person's perspective on the topic, and while I am a little disappointed that my idea wasn't entirely unique, I think the fact that I'm not the first person to come up with it does lend it some further merit.

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