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:

- Effective HP versus: d20
- Effective HP versus: XdY
- Effective HP versus: exploding dice
- Exploding dice: adding standard dice
- Exploding dice: opposed rolls

The right tail of a geometric distribution has the property that every point multiplies chance to hit and EHP by the same ratio. …

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.

Opposed geometric rolls result in a discrete version of a Laplace distribution, which is basically two geometric distributions glued back-to-back. The half-life on each side is the same as that of the individual geometric distributions. With an opposed roll both players have a chance to succeed or fail regardless of the disparity in modifiers. This is unlike the non-opposed roll, where the roller may be fail-proof if their…

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.

To the first point, the geometric distribution has exactly one parameter, and that parameter directly scales its width. There’s several ways of expressing this parameter, but I’m going to go with the amount of attack bonus needed to halve the opponent’s EHP (as long as we aren’t already at 100% hit…

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.

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 in this series we are only analyzing binary succeed/fail outcomes). In short, if we roll the highest number on the die, we roll the die again and add it to the…

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)?

I picked 5d12–22 because it behaves similarly to 1d20 around the median. Here’s what I mean when I say the medians are similar:

Both graphs look about the same near the 50%¹ mark of the “At Least” graph, i.e. the…

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 bad. If this level of “swinginess” is indeed bad, there’s a simple way that a d20 system could be made less so: if we 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…

Articles about probability in RPG design.