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.

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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:

  1. Effective HP versus: d20
  2. Effective HP versus: XdY
  3. Effective HP versus: exploding dice

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:

  1. Using a single relabeled exploding d20. …

Previous parts:

  1. Effective HP versus: d20
  2. Effective HP versus: XdY

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:

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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.

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