The new rules for D&D 5e (formerly known as D&D Next) are finally here:

D&D 5e introduces a new game mechanic, advantage and disadvantage.

Basic d20 Rules

Usually, players roll a 20-sided die (d20) to resolve everyting from attempts at diplomacy to hitting someone with a sword. Each thing a player tries to do has a difficulty and rolling greater than or equal to the difficulty (with various modifiers for ability and training and magic items) means the character was successful.

Advantage and Disadvantage

As of 5th Edition (5e) rolls can be made with advantage or disadvantage. The rules are:

Advantage: roll two d20 and take the max

Normal: roll one d20 and take the result

Disadvantage: roll two d20 and take the min

So what are the chances that you’ll roll equal to or above given number with advantage, normally, or with disadvantage? Here’s a table.

roll disadvantage normal advantage 20 0.002 0.050 0.098 19 0.010 0.100 0.191 18 0.022 0.150 0.278 17 0.039 0.200 0.359 16 0.062 0.250 0.437 15 0.089 0.300 0.510 14 0.123 0.350 0.576 13 0.160 0.400 0.639 12 0.202 0.450 0.698 11 0.249 0.500 0.751 10 0.303 0.550 0.798 9 0.361 0.600 0.840 8 0.424 0.650 0.877 7 0.492 0.700 0.910 6 0.564 0.750 0.938 5 0.640 0.800 0.960 4 0.723 0.850 0.978 3 0.811 0.900 0.990 2 0.903 0.950 0.998 1 1.000 1.000 1.000

The effect is huge. There’s less than a 9% chance of rolling 15 or higher with disadvantage, whereas there’s a 30% chance normally and a 51% chance with advantage.

[Update II: Turns out that The Online Dungeon Master, a blog I read regularly for 4e module reviews and play reports, generated the same table over a year ago (using Excel simulations, no less; a commenter provides a good breakdown of the analytic solution). That’s been the story of my life coming into a new field. I agree that this tabular form is what you want for reference, but the easiest way to understand the effect is looking at the non-linearity in the graph below.]

Here’s a plot (apologies for the poor ggplot2() and png() defaults — I don’t understand ggplot2 config well enough to make titles, clean up labels, axes, tick mark labels, boundaries, margins, colors, and so on to make it more readable without spending all night on the project).









The vertical distances at a given horizontal position show you how much of a bonus you get for advantage or disadvantage.

[Update: There’s an alternative plot on the Roles, Rules, and Rolls blog that displays the difference between advantage and a simple +3 bonus, as used in previous D&D editions.]

Analytic Solution

The probabilities involved are simple rank statistics for two uniform discrete variables.

You can compute these probabilities analytically, as I show in Part III of the page where I explain the math and stats behind my simple baseball simulation game, Little Professor Baseball.

The basic game is a cross between All-Star Baseball and Strat-o-Matic. I wanted a rolling system where both players roll, one for the batter and one for the pitcher, and you use card for the player with the highest roll to resolve the outcome. The winning roll is going to have a distribution like the advantage probabilities above (except that Little Professor Baseball uses rolls from 1–1000 rather than 1–20. As a bonus, earlier sections of the math page explains why Strat-o-Matic cards looks so extreme on their own (unlike the All-Star Baseball spinners).

I developed the game after reading Jim Albert’s most excellent book, Curve Ball, and I included the cards for the 1970 Cinncinnatti Reds and Baltimore Orioles (which for trivia, were mine and Andrew’s favorite teams as kids).

Simulation-Based Calculation

I computed the table with a simple Monte Carlo simulation, the R code for which is as follows.