Unlucky: the statistics of Dungeons and Dragons

Like any game with a chance component, Dungeons and Dragons invites superstition. Sooner or later, players develop a preference for lucky dice (or lucky socks). With the COVID-19 pandemic dragging on, my Dungeons and Dragons group has moved its sessions from the tabletop to a popular online platform for “pen and paper” RPGs, and the platform’s random number generator has quickly replaced our Dungeon Master’s dice as the object of our superstition and probabilistic ire.

Three critical fails in one round? The random number generator must have left its entropy at home!

• Me, circa Sunday afternoon

I often find myself wondering “What are the chances?” after a particularly bad roll of the dice. Recently, that got me thinking: what are the chances?

Funny shaped dice and probability distributions

For those who don’t play, a lot of Dungeons and Dragons revolves around rolling strangely-shaped dice to determine whether difficult actions are successful, and sometimes how much of an effect the action has. In simple cases, only one die is rolled, and the value on the side of the die determines the outcome. In more complicated situations, several dice are rolled at once and the total value across all of the dice is used. Of course, when only one die is rolled, any outcome is equally likely --- in statistical language, the value of the roll is a random variable taken from a uniform (flat) distribution. However, when multiple dice are rolled, some outcomes are more likely than others because there’s often more than one possible roll that will add up to a particular value.

For example, consider a roll of two four-sided dice with the sides numbered one to four. There’s only one way to get a total roll of two: if both dice land on one. On the other hand, there are two ways to get a total roll of three: the first die could land on one and the second could land on two, or the reverse could happen. Continuing the pattern, there are three ways to get four, four ways to get five, three ways to get six, two ways to get seven, and one way to get eight. Clearly, all outcomes aren’t equally likely, and the probability distribution over possible outcomes is far from flat. Things quickly get worse as we add more sides or more dice.

So, what is the shape of the probability distribution over values of a roll of multiple dice? It’s complicated. Check out the widget below to see how the number and type of dice being rolled affects the shape of the distribution. (If you happen to be in the middle of a game of Dungeons and Dragons, you can also mouse over the chart to find out just how bad your last roll actually was.)

There is a 62.5% probability of getting 5 or lower.

Two useful results from probability theory

If you’ve spent a bit of time playing with the widget above, you’ve probably noticed a couple of things.

1. The distribution of a sum of independent random variables is convoluted. Literally. It’s the convolution of the underlying probability density or probability mass functions. Convolution involves sliding two functions past eachother and calculating the size of the area of overlap. This explains why the shape of the probability distribution for a roll of any two dice with an equal number of sides has a triangular shape, for example. (It also happens to be how the probabilities in the chart above are calculated.)
2. As we add up more random variables, things start to look a bit more normal. Literally. The distribution of a sum of independent and identically distributed random variables gets closer and closer to a Normal distribution as more variables are summed. This is called the Central Limit Theorem. This explains why the probability distributions of rolls with several dice with the same number of sides are shaped like a bell curve[^1].

[^1]: Although the Central Limit Theorem technically only applies to identically distributed random variables, notice that the values of rolls involving dice with different numbers of sides also follow roughly bell-shaped distributions. This illustrates the robustness of the Central Limit Theorem, and helps explain why the Normal distribution shows up so often in nature.

Sum up

The take-home: if you don’t like uncertainty in Dungeons and Dragons, pick builds that will let you roll more smaller dice over fewer big dice. Stay lucky!