Comparing the dice of D&D and PbtA with Math

A post by alumnus Austin Tinkel got me thinking about dice and probability, and by extension, how these things are often misunderstood even by those for whom rolling dice is an essential part of their avocation. His is a thoughtful post that covers a lot of ground, and I encourage you to read it. My point in writing this article is to address a fallacy that I see in his and many other RPG essays on the Internet, particularly those comparing Dungeons & Dragons to Apocalypse World and the Powered by the Apocalypse (PbtA) movement.

Here is a summary of the difference between these two systems. In the D&D line of games, random outcomes are determined by the roll of a d20 against a target number. For example, you might need to roll a 16 to hit a bandit with your greatsword. In this situation, any roll of 16 or higher is a success and any roll 15 or below is a failure. A character's environment or attributes might alter their role positively or negatively, but we can ignore those modifiers for now without loss of generality. Personally, I am keen on the ICRPG approach in which an entire encounter has a threat rating. For example, an abandoned castle may have a threat rating of 12, meaning that for any roll the players make, they need a 12 or better on a 20-sided die, whether that is to pick a lock, topple an evil shrine, or strike a zombie.

Games that draw on Apocalypse World use a different resolution system. Players roll 2d6 for any situation where there is a chance of failure. On a total of six or less, it is a failure; on 7–9, it is a partial success; and on ten or higher, it is a full success. Note the obvious property that this is a ternary system rather than a binary one. As in D&D, difficulty can be modified by the environment or character attributes, but as before, we can ignore this without loss of generality.

Here is where I see a lot of people make a mathematical mistake. I see fans of PbtA claim that these games are more likely to get partial success outcomes because of the outcome distribution of 2d6. There's a problem with this claim, though: more compared to what? It's true that 2d6 will give you results in the middle of the distribution more often, whereas a d20 will give each result with equal frequency. However, that does not imply that you get partial successes more often with 2d6 because it entirely depends on the target number for the d20 system. Remember: our simplified PbtA fixes the partial success target at 7, whereas the D&D approach leaves it unspecified.

The distinction can be clarified by using (you guessed it) math. It's helpful to switch from thinking about distributions to thinking about probabilities. The advantage that mathematical probabilities give us is that they are actually independent of the distributions that produced them. Consider: whether you are going for a six on a d6 or trying to draw one of 17 white cubes from a bag that contains 83 other colors as well, it's still a 17% chance. That's the beauty of math. (Note that I'll be rounding my percentages to the closest integer for this discussion, since that's all the precision we need.)

On 2d6, there is a 58% chance to roll seven or higher, and so there is a 58% chance to roll some kind of success in PbtA. This breaks down into a 42% chance of a partial success and a 17% chance of a full success. In D&D, you could get basically the same chance of success by setting the target number for a roll to 10, since there is a 55% chance to roll a ten or higher on a d20. The minor differences between percentages (58% to 55%) is certainly below the threshold that any player would notice the difference. Remember that the pertinent difference between the two systems is not in the dice themselves but the fact that D&D has binary outcomes while PbtA has ternary. If you wanted to set up a situation in D&D that mimics the mechanical structure of PbtA, set it up so that 10-17 is a partial success an 18-20 is a full success. This lacks some of the beauty of PbtA dice that I wrote about last year; namely, that 7 and 10 are memorable threshold values in a way that 18 seems arbitrary. It is important to note, though, that you end up with a mathematically equivalent system, because chance of success is a matter of probability and not distribution. Whether you want to let 20 still be a critical hit in D&D—or let 12 be a critical hit in PbtA—can be a topic for future religious wars.

It is interesting to turn our attention to the effect of bonuses in either game. In PbtA, if you have a +1 to a roll, your chance of getting at least a partial success jumps from 58% to 72%. Interestingly, because of the distribution, each successive bonus provides diminishing returns: the first +1 gives a difference of about 14%, while successive bonuses give 11%, 8%, 6%, and 3%, until you hit 100% chance of success. In a d20-based system, of course each bonus gives a flat +5% bonus. It seems like this should be a significant difference, whether your smallest possible bonus gives a +11% or a +5% boost. However, I suspect that our human brains are so bad at statistics that, in practice, the difference is undetectable by human experience. Another way to put that is that the psychology of a bonus is probably much more important than the distinction between perceived effects. This is only a hypothesis, however, and I think it would make an interesting study.

There is an important corollary to the previous point about bonuses. In a d20-based implementation of ternary results, you can move the goalposts more subtly than you can in a 2d6 system. In the same way that ICRPG has you set a target number for the whole ruined castle encounter, one could independently set the partial success target and the full success target in order to propel the narrative. I can imagine a pair of oversized dice to indicate each boundary value, starting at 10 and 18. When the players knock over the evil shrine, they release a curse that reduces the chances of anything going well, so the 18 becomes a 19: more partial successes, fewer full successes. Most of the time, that 5% difference would go unnoticed, but it becomes like a critical hit in traditional D&D. When someone rolls the boundary value, there will be excitement and fear.

That sounds a bit fiddly to me, and I would just play Dungeon World if I wanted a PbtA-style fantasy adventure rather than hacking the d20 system. However, if you have tried playing with variable outcomes in a D&D game, I'd love to hear from you.

I used AnyDice.com to help with the probability calculations for this post. It's a nice tool that I wish I had when I was working on my too-many-dice RPG in the 1990s.