Different distributions do have different probabilities of success. e.g. Rolling 15 or higher on 1d20 has an 30% chance of success vs Rolling 5 or higher on 3d6 has a 9.26% chance of success vs Rolling 5 or higher on 1d100 has a 86% chance of success.

But these aren’t comparing the same thing.

A task that is “difficult to complete” in a 3d6 system, requiring 15 or higher, is equivalent to 92 or higher in a d100 system (or a 19 or higher in a d20 system).

Yes, a task that is “average to complete”, e.g. 51 or higher on d100, will have a higher variance than “difficult to complete”, e.g. 92 or higher on d100.

output d100 >= 51 named “51/d100 success distribution” \ p=0.50, sd=0.50 \

output d100 >= 92 named “92/d100 success distribution” \ p=0.09, sd=0.29 \

But that has nothing to do with the dice rolled, but with it being closer to a 50/50 chance.

Whether a neophyte has a 50% chance of failure vs a trained character having 25% chance of failure is entirely based on how the system defines neophyte vs trained, and nothing to do with the dice used.

]]>It is only not less swingy for tasks that succeed or fail around the median. For tasks that one has a good chance pf success at (say need a 5 or higher), 3d6 gives a far greater chance of success than a d20, and for difficult tasks (require a 15 or higher), have a far less chance of success.

It is these outliers that have designers cry “too swingy” for d20s. Why would your character who has trained for years, have a 25% chance of failure at getting a 6, and a neophyte still has only a 50% of failure at the same task?

]]>The multiple size d6 pool roll under mechanic seems a little complicated, with varying dice pool sizes, roll under, compare highest, plus special rules for 1’s & 6’s (in most cases irrelevant), but an interesting way to preserve 50/50 chance between two arbitrarily high similar scores, while preserving differences at lower scores.

Most systems have trouble comparing both 8 vs 9 and 80 vs 90; using a different dice range would be one approach.

Ignoring details (like all 1’s, ties, what if you both fail, etc), the probabilities can be calculated by breaking the rolls at the shared (lower) range where the probabilities are 50/50.

e.g. For Str 25 vs Str 30 with 6d6.

- A < 25, B < 25 is a 50/50 chance of who wins.
- A < 25, 25 <= B < 30, B wins because they are higher than A can possibly get.
- A < 25, B >= 30, A wins.
- A >= 25, B < 30, B wins.
- A >= 25, B >= 30, both fail

You can get the probabilties from AnyDice `output 6d6`

This is easiest layed out in a table:

A \ B | < 25 (79%) | < 30 (19%) | >= 30 (2%) |

< 25 (79%) | 2x 31% each | 15% B | 1.5% A |

>= 25 (21%) | 17% B | 4% B | 0.5% neither |

A (Str 30) has a 67% chance to win, B (Str 25) has a 32.5% chance, so about 2/3 vs 1/3 chance.

In contrast with a Str 8 hobbit, the chance of rolling less than 8 on 6d6 is 0.02%, so their chance of winning is almost zero.

The upper end is always small (as the dice changes to be just higher than the maximum), so the key is the lower scored participant gets 50% of the shared chance (which is the lower chance vs the entire dice range).

**Example 1:** Str 15 vs Str 20, with 4d6, the shared (< 15) is 66% chance, of 0.56 * 0.56 * 0.5 ~ 16% chance for Str 15 to win.

**Example 2:** Compared to Str 55 vs Str 60, with 11d6, where the chance is 0.998 * 0.998 * 0.5 ~ 50%; the chance of both rolling under 45 on 9d6 is almost certain, so it is a straight 50/50 contest.

So, if the lower score is about half the maximum dice range, the shared chance will be about 50%, giving the lower score a 25% chance to win. Lower scores will be less than 25%, and higher scores up to 50% (both non-linear changes).

A lot depends on what the system is trying to achieve, e.g. in D&D, a game of heroic fantasy where individual heroes can defeat mighty monsters, they use bounded accuracy with a linear systems: the halfling hero could very well have Strength 20, and the cloud giant only Strength 27, so comparable, especially if the halfling has proficiency in athletics. This is the kind of movie / heroic action scene, and as everything is kept within a small range, comparing two giants is also easy.

The other end is having a scale system, e.g. Savage Worlds heavy armor and heavy damage, where normal weapons simply can’t hurt tanks (at all) and two tanks just compare normal weapon ratings. Cortex Prime has something similar with Scale Die, but also allows heroics (as characters can spend plot points to include extra dice).

]]>Michael,

Whoa there cowboy! First off lets go easy on the mathematically challenged. I think the reason most people don’t get the underlying math is that in the real world, (home, jobs, daily life), the vast majority of people don’t even use it. Thank the education system, calculators and the industrialization for that. Henry Ford doesn’t need mathematicians to tighten down the seat, Amazon isn’t hiring statisticians to deliver packages. Common Core is designed to provide workers, not thinkers. On the other hand, do you know how to make lavened bread from soil and seed? Properly temper a knife you hammered yourself in a homemade coal forge? Unlikely because they sell Wonder Bread and knives so inexpensively. : ) Nuff said.

On to the real discussion. You stated:

‘The only way to attain believable results is to use decimal multipliers, like x0.8’

Hum..???… I think your right.

If I remember correctly CoC 7ed and Mythras attempts to address the -20% ‘high meaningless, low skill massive’ by tests which are a fraction or multiple of your base skill. Like hard being 1/2 skill so a 50% skill becomes 25% and a 20% skill becomes 10%. What I don’t like how it penalizes high skill characters much more than low skill characters. A master with 100% who would normally succeed 9.5 out of 10, (96-00 always fails) now fails 5 out of 10 times while a stooge, (20%), who normally fails 8 out of 10 times now fails 9 out of 10 times.

Your x0.8 solves this but becomes math intensive at the table. “You have 77% skill with a x0.8 mod for range, x0.5 mod for cover and x1.5 mod for the ‘seeker’ spell on the arrow.

I am not good enough at math to think of an easy way to overcome these multiple mods in play. Any ideas?

So the bell curve of 3d6 does seem to work better for mods than a % system. What you lose is granularity and steady skill progression. Each point of skill gained is now referenced to your current skill. 1 point will have a value 0.46 to 12.5. A single ‘bit’ of information learned by someone in the middle, (such as the names of a couple demons for a sorcery skill), might increase his skill chance by 0.46%, (2 skill becoming 3 skill using 3d6), or 12.5%, (10 skill becoming 11 skill using 3d6). It is also far less obvious that if I have a 12 skill on 3d6 that it equals a 74% chance than 74 Skill equaling 74% using a percentile roll. I believe we tend to think in ‘slices of pie’ more than bell curves.

Perhaps skills increasing as percentages yet rolled on 3d6 to the nearest percentage would work better when modifiers are applied? Either that or still rolled on percentile dies but your fractional modifiers would be charted out for ease of use. That’s gonna be a big chart to cover all the 100 skill levels and multiple mods.

I have also though about things like Stat Tests being based upon the highest Stat in the contest rather than a static 3d6 under roll or a Resistance Table, (Basic Roleplaying), where each point of difference equals a 5% modifier.

1-5 = 1d6, 6-11 = 2d6, 12-17 = 3d6, 18-23 = 4d6, etc; Where the the highest Str is always at least 1 point above the maximum die roll. All 1’s is always a success and all 6’s always a failure. Thus when when a giant with Str 30 is arm wrestling a Hobbit with Str 8 they both roll 6d6. Winner is the one who rolls highest but also under their Str. Its not so much for the contests of 8 vs 30 as it is for 25 vs 30. A mechanism where each point is a 5% modifier means a 6 vs 12 is the same as a 12 vs 18, 20 vs 26 or a 100 vs 106. This is fine IF each point is significantly higher than the previous in terms of its value where a 106 Str carries twice as much weight as a 100 Str, (being equivalent to 6 vs 12 or 12 vs 18), but fails if the points are of equal value. Also, the mechanism of dding the result to a die roll such as 1d20 + Str where the highest result wins is fine when a 6 vs 12 but not when a 20 vs 39 or 100 vs 106.

This came up when I was designing a d100 campaign which revolved around warring giants of various clans and types and realized the mechanism for resolving Stat vs Stat battles was not very good.

Any thoughts on this?

Thanks for your insights and input!

Mike

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