Wednesday, May 9, 2018

approachable theory: Tabletop RPG Dice Math

The approachable theory logo, with the text "approachable theory" and an image of two six-sided dice with one pip showing, with a curved line below it to make a smile. The dice are black with cyan for the pip and yellow with black for the pip.
approachable theory logo. (By Brie Sheldon)
Hi all! I have a post today from Michael "Karrius" Mazur (email) about tabletop RPG dice math. Michael is a tabletop RPG player, more often a GM than not, and in his own words, he's "always had an interest in tinkering with and designing game systems." I asked him his favorite part about roleplaying games and he said it was that roleplaying games are a creative hobby with a low barrier to entry - and I like that too! 

This post is definitely a lot of information, but I think Michael explains it simply and approachably. Please enjoy!


Mathematical balancing can be an intimidating subject for RPG designers, but crucial for making a game that works like the designer intends is the solid foundation of having the correct dice rolling method. While familiarity with some methods of rolling are usually understood by designers, what questions to ask when deciding on a dice rolling system, the differences between said systems, and how to pick what’s most suited to the game can be a tough subject. The first question to tackle is often one of how many dice are appropriate.

The most familiar method of rolling dice in RPGs is the simple “roll a 1d20, add a number to it, and compare to a target number,” due to its use in Dungeons and Dragons and related spin-offs. One common modifier to that is instead of rolling 1d20, use the sum of 3d6. These two methods make for useful comparisons of the difference between rolling a single die vs rolling multiple dice and adding them. Both average out to the same result (10.5), and have similar maximums (20 vs 18) and minimums (1 vs 3). The following graph shows the probability curve of rolling each result on a 1d20 and 3d6, comparatively.

A graph showing the probability of rolling an individual result on 1d20 or 3d6. The 1d20 odds are a flat line, with 5% chance of rolling any number. The 3d6 results are a bell curve, as high as 12.5% for 10 and 11, and as low as 0.5% for 3 and 18.
This chart shows the probability of rolling a specific result, visual by Michael. Full details in alt text.
The probability of getting any specific result on a 1d20 is equal - each number is equally represented by once face of the die, giving a 5% chance of rolling any number in its range. The flat nature makes it easy to do calculations with - if you have something that activates on certain numbers being rolled (like critical hits on 20, or special moves on even rolls), it’s easy to just add up those 5%s, even in your head at the table. On the graph “Probability of Rolling a Specific Result,” the line for the 3d6 is a “bell curve” shape - the most common results are in the middle of the dice range, due to an “averaging out” effect, where there’s multiple different sums that can achieve them.

Rolling the maximum or minimum are far less likely due to having fewer combinations of dice that can achieve them. While a 1d20 can be expected to roll within a half point of the average result (10-11) one time in ten, and roll the maximum result one time in twenty, a 3d6 rolls within a half point of the average an expected one time in four, and roll the maximum result one roll out of every 216. Rolling multiple dice causes the “typical” results to be far more likely, and the extremes to be far less so.

click through for more!

The next step on the analysis is to understand that when you’re playing the game, for most rolls you’re not looking to roll an exact result - you’re looking to roll that number or higher - to succeed on the difficulty of a task. The following graph shows what the probability is of succeeding on a task for at various results needed, comparing 1d20 to 3d6.

A graph showing the probability of rolling a specific number or greater on a 1d20 or 3d6. The 1d20 odds are a diagonal line, with a 5% step each increment. The 3d6 results are a curve, with greater odds of success compared to 1d20 for results below 11, and lower odds of success for results above 11.
This chart displays the probability of rolling a specific result or greater, visual by Michael. Full details in the alt text.
What does it mean for the 3d6 that the extreme results are less likely, and average results more common? The biggest take away is simple: that if you’re already favored - and so the average result is a success for you - using 3d6 instead of 1d20 means you become even more so. But if you’re relying on an extreme result to have a chance of success, moving from 1d20 to 3d6 lowers your chance of success even more. Using a 3d6 allows a character to perform “lesser” tasks reliably, but struggle at performing tasks above their skill level. It promises easy success to the specialized, and warns away those not well suited to a task from even attempting it.

Situations where a skilled character repeatedly fails at a given task over and over become far less likely, which can ease frustration or keep a challenging monster fight from becoming trivial due to a few bad rolls from the monster. Systems that roll 1d20 allow a wider variety of task difficulties, which become possible for characters to attempt sooner, but continue to have a noticeable failure rate as a character grows. Rolling 1d20 gives a character less reliability, but also more leeway if they’re being forced into attempting more difficult tasks.

Deciding which tasks - reliable or risky - you want your dice system to encourage isn’t the only change in player behavior you’ll see from it either. An important consideration is how valuable a +1 bonus is to the success at any given task. This won’t be obvious to all players, but some will understand how this works, and others will pick up a sense of it as you play. The following graph shows the probability of a +1 bonus changing a failure to a success for each roll needed.

A graph showing the probability of a +1 bonus to a roll causing a success depending on what number is needed to be rolled for 1d20 or 3d6. The graph is the same shape as the odds of an individual result on 1d20 or 3d6. The 1d20 odds are a flat line, with 5% chance of rolling any number. The 3d6 results are a bell curve, as high as 12.5% for 10 and 11, and as low as 0.5% for 3 and 18.
This chart shows the probability of a +1 helping, visual done by Michael. Full details in the alt text.
The probability of a +1 bonus having changed a failure to a success is the exact same as the probability of rolling the number one fewer than whatever you’re trying to roll. As such, the graph of the +1 bonus helping is the same as the graph of individual roll results, except shifted one number higher. If you’re rolling a 1d20, no matter what task you’re performing, a +1 bonus is giving a 5% greater chance of success. If you’re rolling 3d6, that instead varies.

For tasks which are near impossible or very easy to perform, the +1 might not be worth sacrificing much for. But for tasks that you already have a near-even chance at, the value is very high - over twice as much as you would be getting on a d20! Players are more incentivized to spend resources or time in order to get any small bonus they can when they expect they’re close to the average of an attack or defense.

All of the above shows the difference between 1d20 and 3d6 - but what about other dice, or rolling even more? The following graph shows a comparison between six types of dice rolling methods, all with roughly even averages of 10-11.

A graph showing the probability of rolling an individual result comparing between 1d20, 2d10, 3d6, 4d4, 5d3, and 7d2. The more dice are rolled, the higher the likelihood of rolling the average results are, and the steeper the “bell curve” shape becomes.
This chart shows the probability of rolling a specific result, visual done by Michael. Full details in the alt text.
The more dice are rolled, the steeper the bell curve becomes. Anything that could be said about 3d6 in comparison to 1d20 is even more so for 4d4, and the trend continues to strengthen the more dice are rolled. If you want it so the average result comes up very often, and people with even small advantages are heavily favored, you want a system that rolls many dice at once. Another design consideration that emerges is the range of values possibly - when rolling 1d20, there are 20 possible results. When rolling 3d6, there are only 16 results - and four numbers (9-12) come up almost half of the time. When rolling 7d2, there are only 8 possible results, and two numbers (10-11) come up 55% of the time.

When so few numbers are likely to be rolled, this means that characters gaining small bonuses or penalties can very quickly put them in the realm of instant success or failure. It only takes a +4 to go from the average value to an instant success on a 7d2, as opposed to a +10 on a d20, meaning there’s not a lot of room for individual bonuses to make a difference and still have a chance of failure. This however also allows a stratification in difficulty - it doesn’t require very large bonuses for an expert character to be able to complete tasks a non-trained character cannot, keeping numbers reasonable if you want multiple tiers of difficulty.

In summary, there are multiple considerations that go into choosing the dice rolling method for your game. A single large die, or fewer dice in general, is more appropriate if you want a game to have multiple conditional sources of bonuses such as gear, positioning, and teamwork, to keep tasks from ever seeming too reliable, and to encourage risky maneuvers. Multiple dice, or smaller dice, are more appropriate if you want to make small bonuses matter a great deal, allow characters to treat lower-difficulty tasks as trivial, or to have hard tiers in difficulty. Neither method is inherently superior; instead, both are proper tools for their tasks.


Thanks so much to Michael for the excellent article! Please share around and I hope you all gained something from reading this post on approachable theory!

If you would like to write an approachable theory post, send an email to Brie with your name, pronouns, and pitch. Responses may be delayed over the next two weeks as Brie is recovering from grad school, but they'll get back as they can.

Updated 5/9/18 12:55pm Eastern to change "odds" to "probability." Failure on Brie's part to not catch that mathematical terminology difference. Sorry!

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