With approximations to geometric dice in hand, we now ask: who rolls them, and how many do they roll?
One player, one geometric roll
Unlike the uniform and normal distributions, the geometric distribution is not symmetric.
If the player always rolls one (non-opposed) geometric die + modifier, they can never score below their modifier, but there is no upper bound to what they could score. This may give a sort of “heroic” feel: the player always succeeds on things that are sufficiently easy for them, and they have a chance (however tiny) to succeed no matter how difficult the task unless it’s flat-out impossible.
Opposed geometric rolls
Previous parts:
Three schemes
Last time, we looked at exploding dice and canonical geometric dice. The former was not a great approximation to a geometric distribution, while the latter was too time-consuming to roll.
Can we find a compromise that gets closer to a geometric distribution but has better ergonomics than the canonical geometric die? Here I present three schemes for approximating a geometric distribution:
- Using a single relabeled exploding d20. …
Previous parts:
As we saw last time, the tail of a normal distribution falls approximately as exp(-x²), which makes the EHP in the right tail grows approximately as exp(x²) (up to some scaling). A gentler curve would be exp(-x) and exp(x) respectively: a geometric distribution.
Exploding dice
In fact, there is an existing die concept that roughly implements a geometric distribution: the exploding die, as used in games such as Savage Worlds (although here we are only analyzing binary succeed/fail outcomes and not yet analyzing mechanics such as “raises”). In short, if we roll…
Previous: Effective HP versus: 1d20
It’s well-known that as you sum dice together that you start to get a bell curve shape (normal distribution). For standard dice this happens quite quickly; even three d6s are enough to start approximating this shape (program #1 on AnyDice). How does this affect effective hit points compared to a single die (uniform distribution)?
1d20 vs. 5d12–22
Matching the medians. I picked 5d12–22 because it behaves similarly to 1d20 around the median (where the problem wasn’t). Here’s what I mean when I say the medians are similar:
Both graphs look about the same near the 50%¹ mark…
The most famous RPG is Dungeons & Dragons, and its core mechanic is rolling 1d20, adding a modifier, and comparing it to a target number such as Armor Class (AC).
It’s often said that rolls in 5e D&D feels “swingy”, and often in a way that implies that this is bad. If it is indeed bad, there’s a simple way that a d20 system could be made less swingy: if you multiplied all the modifiers by 10, the d20 would mean almost nothing, and the modifiers almost everything. Even a more reasonable multiplier of 1.5 or 2 would make the…
High Dice Roller
Articles about probability in RPG design.
