With tabletop roleplaying game (RPG) systems, sometimes I hear a claim that bell curve dice rolls, e.g. 3d6, are “less swingy” (less variance) than a linear based dice roll such as d20 or d%.

This is, however, incorrect.

While the distribution of dice rolls are different, the distribution of outcomes – success or failure – are the same, and for equivalent circumstances have the same statistical variance / standard deviation.

Although the outcomes have equivalent distributions, the underlying type of dice system is important for analysis of modifiers and skill progression.

## Dice probability distributions

RPG systems use many different resolution mechanics, from simple to complex, from diceless to card based, with various sized standard dice, or sometimes custom dice.

Examples are provided comparing three common types of system, all of which have free downloadable quick start rules if you want a closer look:

- Linear – Call of Cthulhu (https://www.chaosium.com/cthulhu-quickstart/),
- Bell curve – Dragon Age (https://greenronin.com/dragonagerpg/#tab-id-2), and
- Dice pool – Shadowrun (https://www.shadowruntabletop.com/downloads/).

The graphs include the expected value and the standard deviation (“swingyness”). You can verify these dice probability distributions using a tool like AnyDice, https://anydice.com

`output d100 named "output d100 named "Linear d100 distribution"`

output 3d6 named "Bell curve 3d6 distribution"

output [count {5,6} in 8d6] named "Dice pool 8d6 distribution"

## Success by skill level graphs

While the dice distribution is one thing, what is generally relevant is the probability of success for given criteria. As skill goes up, or difficulty goes down, the chance of success increases, and we can show a graph of the cumulative chance of success.

We can graph the chance of success, for a given difficulty (Cthulhu hard test, Dragon Age target number 15, Shadowrun threshold 2), against different skill levels, for each type of dice system:

These three graphs are the three typical patterns for how you see chance of success increase by skill. A linear dice roll gives a linear increasing probability, bell curve gives an S-type shape, and a dice pool has the top half of the S-shape (or inverse logarithmic type shape).

Note that these are not probability distributions, but the probability of success under different circumstances.

## Success probability distributions

For any given test, the test is a certain difficulty, and the character has a particular skill level, and we can determine the chance of success or failure outcome.

We can graph the probability distribution of outcome results, as well as calculate the average and standard deviation, for each of the different dice systems.

- Linear: Call of Cthulhu, skill 52% hard test (half skill).
- Bell curve: Dragon Age, ability + focus 2 vs hard test (target number 15).
- Dice pool: Shadowrun, attribute + skill 8d6 vs hard test (threshold 4).

These can be generated in AnyDice, including the statistical values, using the following:

`output d100 <= 52/2 named "Linear success distribution" `

output 3d6 + 2 >= 15 named "Bell curve success distribution"

output [count {5,6} in 8d6] >= 4 named "Dice pool success distribution"

The distribution of the results is binomial, and in equivalent circumstances has the same distribution, irrespective of the distribution of the underlying dice roll system.

### Multiple tests

Where there are multiple tests, such as a combat sequence, where the tests are equivalent, then the results (how many times you succeed), follow that binomial distribution and are also equivalent, irrespective of the underlying dice mechanic.

To verify the different dice systems producing the same outcome probability distribution in AnyDice:

`function: five X { result: X+X+X+X+X }`

output [five d100 <= 52/2] named "Linear multiple success distribution"

output [five 3d6 + 2 >= 15] named "Bell curve multiple success distribution"

output [five [count {5,6} in 8d6] >= 4] named "Dice pool multiple success distribution"

### Effects of a specific modifier

The effects of an equivalent modifier likewise produce equivalent binomial distributions.

- Linear: Call of Cthulhu, skill 52% hard test (half skill), with 2 bonus dice
- Bell curve: Dragon Age, ability + focus 2 vs hard test (target number 15), with a +3 modifier.
- Dice pool: Shadowrun, attribute + skill 8d6 vs hard test (threshold 4), with +4 bonus dice.

To check these modifiers in AnyDice:

`output 3@3d{0..9} * 10 + 1d10 <= 52/2 named "Linear with modifier success distribution"`

output 3d6 + 2 + 3 >= 15 named "Bell curve with modifier success distribution"

output [count {5,6} in (8+4)d6] >= 4 named "Dice pool with modifier success distribution"

### Distributions are the same

Equivalent graphs have been shown three times in a row, to emphasise the point that all three systems have the same binomial distribution of outcomes (success or failure), even though the underlying dice systems are different.

Using 3d6 is not less (or more) swingy than d100 due to the dice, although the probabilities will of course vary due to other differences in the systems, such as how skills or modifiers are calculated.

## So, how is the dice system relevant?

While it doesn’t have an impact on the probability distribution of success outcome, the dice system does have an effect on how modifiers vary, skill progression, and on the margin of success.

### Skill progression

An important tool for analysing a game system – for different difficulties (easy/average/hard) how often do you want characters to succeed across varying level of skill, given the style of the game.

These can be used to analyse if tests in the system are too easy or too hard, or guide the gamemaster’s choice of difficulty. e.g. even an expert is going to fail more than half the time for a hard test in Call of Cthulhu, whereas even a novice has a fair chance of success for an average test in Dragon Age.

The graphs can also be used to detect any anomalies, where a higher skill has a lower chance of success. An example is Savage Worlds where against a target number of 6, a skill of d4 has a better chance of success than a skill of d6.

Note these graphs can’t be created directly in AnyDice. The best you can do is use a Transposed Graph to show the range of success (and failure) values for a single difficulty.

loop P over {1,10,20,30,40,50,60,70,80,90,100} { output d100 <= P named "[P]" }

Make sure you use View = Graph and Data = Transposed to make sense of the data.

### How modifiers vary

Rather than look at different difficulty levels, you can also look at what effect bonuses and penalties have.

These graphs are most useful when bonuses and penalties use a different mechanic than difficulty. For example, in Call of Cthulhu you get bonus (lowest of) or penalty (highest of) dice for the 10’s placeholder, adding a non-linear effect, and Shadowrun gives you bonus or penalty dice (rather than change the threshold).

For systems with flat modifiers, such as Dragon Age, the graphs for bonuses and penalties, by skill, and by difficulty, are all the same, just shifted left or right.

### Degree of success

While whether you succeed or not is a binomial distribution, the degree of success – how much you succeed by, sometimes called margin of success – can have a variety of distributions.

Where the system has the notion of critical successes or critical failures, can give an idea of how frequently they will occur and if the fits the intended style of the game.

The distribution of degree of success may be quite different from the system used to generate success or failure, e.g. Dungeons and Dragons has a flat d20 for success checks, but damage often has multiple dice and so is bell curve shaped; in contrast Dragon Age uses 3d6 but the “Dragon Die” for size of effect is a flat d6.

## Alternative system design

Any time there is a success/failure outcome it is a binomial system, irrespective of the dice roll or other system (e.g. cards) used. Degree of success modifies this along a single axis, with some games having more complex resolution systems.

Some systems have both success/failure and advantage/disadvantage determined by the roll, leading to combinations where you might succeed but at a cost (success + disadvantage), or fail but with a benefit (failure + advantage).

The Genesys system has custom dice with different symbols for success and advantage, while Powered by the Apocalypse has outcomes for failure, success at a cost, and success.

Another example is Cortex Prime, where an action can succeed or fail, but any 1’s rolled can also be turned into conditions; Shadowrun has a similar system with glitches. Even Dungeons & Dragons has an advanced “Success at a Cost” rule in the Dungeon Master’s Guide, where a roll can both succeed and produce a penalty.

Outside of random numerical resolution (Fortune), game systems can use straight ability comparison or point spending systems (Karma), or a negotiated story-based result (Drama).

For example, in Gumshoe you either have an investigative skill or don’t; and in the old Amber RPG the highest statistic always wins. Drama based systems range from escalating negotiated statements in Polaris: Chivalric Tragedy at the Utmost North to the pictorial card deck used for subjective outcome determination in the Everway RPG.