Randomizers

As every gamer knows, we generally use randomizers within our games. As regular readers of this blog know, I have a working knowledge of probability. As such, I figured I’d give a brief non-technical primer of types of probability distributions and how to approximate them at the table. Some of this is widely understood in the community and I’ve discussed some of it before, but thought it might help designers and GMs to have it all in one place.

There are a few properties you may want your randomizers to have. The first two are gamer slang and the later two are probability terms that don’t have gamer slang synonyms.

• Swinginess — How unpredictable do you want the outcome to be? Who wins at arm wrestling should probably be less swingy than who wins at rock-paper-scissors.
• Granularity — How many intervals do you want the outcome to have?
• Skew — Should the distribution be asymmetrical, usually because low values are common but extremely high rate values are still possible (“right skew”). In real life, right skewed distributions describe issues like how many, how powerful, or how long.
• Replacement — Do outcomes repeat?

Now let’s talk mechanics and think about what properties they have.

• Cards — Sampling without replacement. If you don’t want treasure table items or encounters or NPC descriptions to repeat, put them on cards, draw cards, and then don’t replace the cards in the deck.
• Dice (whether interpreted directly like attack rolls or used to generate results from a table) — Sampling with replacement. Each roll of the die is independent of the next so you can get the same result repeatedly.
• Small dice, like d6 — Low granularity. Gumshoe uses a d6 for everything so point spends meaningfully affect each “general ability” roll.
• Large dice, like d20 or d100 — High granularity. D&D and Basic Roleplaying (Call of Cthulhu) use large dice so you can stack tiny bonuses. You can upgrade from chainmail to plate, or a +1 to a +2 sword, or 42% to 47% in library use.
• Summed dice — Low swinginess. Summing dice gives you a bell curve with moderate values being more common than very low or very high values. Summing many small dice (eg, 4d6) will be less swingy than summing a few large dice (eg 2d12).
• Roll under ability — Relatively low swinginess. A character with ability 18 will roll at or under that ability 90% of the time and a character with ability 3 will roll at or under 15% of the time. The competent will usually be successful and the incompetent unsuccessful.
• Roll over difficulty with mod — Relatively high swinginess. For difficulty 12, a character with ability 18 and a +4 mod will beat difficulty 12 60% of the time. His teammate with ability 3 and -4 mod will do so 25% of the time. The competent have only a relatively small advantage over the incompetent.
• Exploding dice (ie, roll again on max value) — a right-skewed distribution, but kind of a funky one and with weird properties like exploding d4 has a higher mean than exploding d6.
• Roll 2, keep low — A right skewed distribution. Good model for how long things take, how many or how much, how powerful, etc.
• Roll 2, keep high — A left skewed distribution. Left skew is pretty rare in nature though obviously “advantage” is a big hit as a game mechanic.
• Roll 3+, drop low — A slightly left-skewed distribution. Still more or less bell-shaped but with more high and fewer low values.