I’ve decided to migrate the canonical version of articles to a GitLab wiki. You can also get updates at my WordPress. Mostly this is to get built-in LaTeX support, without which it’s not easy to display math formulas.

Miscellaneous items that I didn’t cover in the series proper, which I tried to keep focused on effective hit points. This part is naturally less polished, being composed of exactly the odds and ends that were left on the cutting board.

The series proper:

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

Last time, we added standard dice to exploding dice in order to smooth out the resulting probability distribution. Now we’ll finish up our discussion of exploding dice by looking at opposed rolls.

The Laplace distribution

Opposed geometric rolls result in a discrete version of a Laplace distribution, which is basically two geometric distributions…

Last time, we looked at exploding dice and canonical geometric dice, each of which had its problems. Our goal is to find a scheme that:

  • Allows us to control the width of our distribution.
  • Stays as close to standard dice as possible.
  • Isn’t too complicated or time-consuming to roll.

Choosing a half-life


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…

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). …

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 feel “swingy”, and often in a way that implies that this is…

High Dice Roller

Articles about probability in RPG design.

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store