Wednesday, June 24, 2015

Determining probability in pen-and-paper RPGs: d20 v 3d6

So back in early 2009 I began looking at the possibility of creating my own Pen and Paper (PnP) roleplaying system, which I dubbed ENSEMBLE. The game system never fully came to life, I started experimenting with Fate and eventually Fate Core, but I had fun at the time.


I was looking for something that could fully function in any setting (or a multiverse setting) without drastic changes to the rules. I had also just experimented with d20 Modern and was dissatisfied with the system as a whole, so I began experimenting with different gameplay mechanics. I had read somewhere that 3d6 offers a better representation of real-life chance than a single d20, so I wrote the piece below in a wiki I had for my burgeoning game system. I stumbled upon that wiki recently, and thought this was mildly interesting and worth sharing. Here we go...

Note: I'm not crazy (really! =P). This reads somewhat conversational because my brothers were posting comments and notes alongside this and other posts on the wiki.

d20 vs. 3d6


Ok, so here's the deal. For quite some time now I've been considering the pros/cons of utilizing a 3d6 system vs. the d20 system… I have a feeling that the initial reaction may be opposition, but hear me out. I'm trying to decide if its really worth it…


See, the benefit of a d20 system is simplicity. Its one die to roll, the bonuses are universal (statistically) and the math is straight forward enough that you can do it in your head. However, its downsides are linear probability that results in unrealistic outcomes in every event and significantly more effort required to balance the system between archetypes. This is the reason that the DnD has the "take 10" rule. Any roll you make is just as likely to turn out a 1 as a 20, so a smart player will "take 10" whenever possible, to avoid the 9/20 chance that they'll fail a roll.

Whereas, a 3d6 system (e.g. GURPS) relies on Normal Distribution, which produces a bell curve. The result being that the mean is more probable than the min/max. This type of modeling more closely resembles real world behavior (as noted in the wikipedia article.) There are many benefits to using this kind of system, the first and foremost is definitely the realism. Having a system where outcomes are statistically realistic allows for more balanced gameplay, as a less-experienced character will be more likely to fail at an event than an experienced character (see charts below). In addition, this facilitates new content creation, as less concern needs to go into balancing each new ability between varied characters. Also, there's the immediate benefit of accessible game tools. Six sided dice are readily available and a new player can find a set at his local gas station. An ancilliary benefit is the slight mystification of the odds. The average player will understand intuitively that as he progresses he will succeed more often, but wont know precisely his chances of success on any particular roll (unless you're Musil =P)

The downsides to a 3d6 system, lie in the personal preference of the player. For example, its just as easy to roll three dice and add your modifiers as it is one die, however the perception is that one die is simpler. In addition, a 3d6 system would likely utilize a "roll-under" mechanic, as opposed to the more common "roll-over" mechanic (as you would want a higher percentage of success vs. high stats) (After additional research I can't say this is true anymore). Also, to utilize a 3d6 system might require some adjustments in thinking, as an attribute score of 15 is significantly better in a 3d6 system than in a d20 system (the odds of rolling under 15 are higher in a 3d6 system, see charts below). Players familiar with a d20 system may be confused by these changes.

So to summarize, and then to present the graphs. the run-down is like-a-so…

single-die system:


  • Pro's
    • Seemingly More Simple
    • Statistically Universal Bonuses (+5 provides the same benefit to a young, or old character)
    • Simple Probabilities
  • Con's
    • Unrealistic Probability
    • Increases the difficulty of balancing the system, as regardless of skill, the odds are always 1/20.

3d6 system:

  • Pro's
    • Normal Distribution of Probability (They call it "normal" cos it occurs so often in nature). Realism.
    • Accessibility. I'm willing to bet that every house in America has at least 3 six sided dice laying around.
    • Experienced characters will always have better odds for success than newer characters. (i.e. Balance)
  • Con's
    • Arguably Less Simple
    • Less transparent odds.

fig 1. Comparative probability of rolling 1d20 vs. 3d6 with no modifiers

Ok, so here are the graphs. On the first, you'll see the comparative probability of rolling 1d20 vs. 3d6, with no modifiers… on the second, the same dice, rolled with a +4 bonus compared to unmodified rolls… Its important to note that you are comparing Green vs. Red and Blue vs. Orange here. So for our examples above, where we compare a target number of 15 in both systems, you'll see that in a 3d6 system it'll be harder to roll extremely high numbers (making it much more significant an event), but much easier to succeed at low target numbers… This makes it much more difficult for a veteran character to randomly fail at something in which they are skilled..

fig 2. Comparative probability of rolling 1d20 vs. 3d6 with plus 4 modifier

Also, the third graph is unique.. It shows the probability of rolling over x… Its hard to convey how this graph effects gameplay, b/c it largely depends on how bonuses are applied.. bonuses in a 3d6 system hold much more weight than in a d20 system (as this graph shows).. And note, that none of this takes into account critical successes or failures…

Also, before I leave you to the beauty of my excel graphmanship, consider the optional rules in DnD's Unearthed Arcana where they allow for a change to 3d6, with some slight modifications to system-wide bonuses, etc. So perhaps its possible to utilize 3d6 in a more simple manner than that I touched upon above.

fig 3. Comparative probability of rolling over X with 1d20 vs. 3d6 

sorry if some of that doesn't make sense… it is 3:00am afterall.. haha…

P.S. I just realized that I think the math is wrong on the 3rd graph… I'll have to work on it tomorrow…

P.S.S. No actually, the math is correct, its just that the graph shows conflicting data… the 3d6 data would be viewed for rolling under whereas the d20 represents rolling over … thus its a flawed representation of "success"… I'll make another one tomorrow to replace it.

The problem with the third graph is that the probability of rolling under with d20 remains 1/20 up until the target number, where it drops off to zero (obviously)… so you would see a flat line, like the other graphs. This further justifies my belief that 3d6 is the way to go, as rolling under a target value is much much more realistic (and easier!) than d20.


Many many months later...


I've decided on 3d6, and want to try and implement it game-wide. The reasons are many, but I'll try to cover them below:
  • Normal Distribution of Probability — This makes sooooo many things easier as far as balancing and stats in general go.
  • Every house in America and most houses across the globe have at least three 6-sided dice. If they don't, probably 90% of gas stations do.
  • Adds a sense of wonder to the game mechanic. Most people can't figure out the odds in their head and will rely more heavily on Heroism Points to pad their rolls—this is highly desired.
  • I want to unify the game mechanic to use 3d6 UNIVERSALLY. I'll elaborate more on this in another post, or possibly in a dedicated page, but the idea being that as you raise in level, your bonuses will raise, thus your roll becomes more powerful. Additionally, all weapons damage can be determined with 1d6 - 3d6, with base damage, bonuses and multipliers enhancing the value (a giant plasma railgun on a mecha might have a base damage of 50 + 3d6, for example)
  • It will differentiate us from the d20 system in a big way. This will be good, as it reprieves us of d20's shadow.
  • I'm not aiming for a Shadowrun-style 30d6 dice pool craziness… it should remain 3d6 at max.. bonuses and base values should handle the rest.
I think that basically covers it. There might be other reasons, but I'm tired and semi-distracted.. so I'll review later and append if needed.

Deciding on this is great though, it allows me to move forward and begin fleshing out the combat systems.

Wrap Up


So yeah... I don't think I ever really figured out what bugged me about that 3rd graph, but I did eventually move forward with a 3d6 concept. It worked pretty well for as far as this system developed.

Anyway, it was fun to revisit this old idea and reread my thoughts at the time. I'm far to busy now and never even play PnP games, but maybe one day I'll revisit ENSEMBLE and flesh it out!


Popular Posts