# What is/are the basic mathematical models for skill-check based systems?

It seems that there is some model-agnostic thinking around skill-check based systems, and I'm interested in finding out what designers use to model them.

Ignoring the dice for a moment, and stating things in terms of probabilities, what are the formula's based on? For example, it seems that a common statement underlying these systems is:

"A normal character attempting a generic skill should have a XX% percentage change of success, with that value increasing in an ZZZZ-shaped curve based on complexity of the specific task attempted."

What are common values for XX and ZZZZ? Are there other baseline models?

After having a baseline formulation, other factors are added in, such as "class", "race", "level" and "bonuses/penalties" and "trained skills" and the like. What are the common modifier models? Why are they great/do they suck?

This is all new to me (and exciting!) - since when I DMed decades ago, we made this all up as we went along, re purposing bizarre things like Potion Miscibility tables and numbers PFA: Pulled From "Air". :-) I think if I could understand the basis better, I'll make a better GM out of the gate.

• I am afraid that it's still all made up—only different designers import different assumptions of demographics, different game priorities, and different degrees of stats comprehension. ;) But of course I could be wrong and there is such a school of thought that I haven't encountered yet, hence this is a comment instead of an answer. Nov 5, 2010 at 4:20
• Considering that Wizards of the Coast has changed D&D 4E's default difficulty values table four or five times over the short lifespan of 4E, I think it's safe to say that these models are still completely made up. Over and over again, they've changed the assumptions about what "easy", "medium", and "hard" mean. And different games will have different definitions of these terms, depending on numerous factors, such as whether the story is supposed to heroic or gritty. Nov 5, 2010 at 5:02
• I hope this comment isn’t out-of-line here... This is one of the things that sent me (and mostly kept me) back to basics. In my experience, all the math of such systems generally doesn’t give better results that PFA did. The better DMs with such systems often end up doing all the math and then applying an ad hoc circumstance modifier (or adjusting the difficulty class) that brings the probabilities in line with what they would’ve PFA’d, so they might as well have just PFA’d it. Nov 8, 2010 at 14:01
• It would be interesting to use some "real world" testing of skill check systems. Perhaps Wizards of the Coast has a special facility for testing % success rates for Easy, Moderate, and Difficult skill checks for jumping over chasms, listening to conversations through a door, appraising a coin, etc. ;) Mar 27, 2014 at 16:25

Joe Normal

For this disucussion, Joe is flat average stats for a human. He is competent in whatever it is you're trying, but not to elite nor professional (IE, Master's Level or equivalent) levels.

For Level based games, usually assume level 1.

Exemplars for Illustrative Purposes

I refer to the following, so the precis is presented so you can follow the actual answer portion without doing research...

MegaTraveller

In MegaTraveller, the task system is based upon a roll of 7+ on 2d6 being "Routine" for a person with Skill 1 and +1 from attribute. MT also defines skill 1 as employable, and Skill 3 professional (as in, doctoral or masters degree trained fields should result in skill 3).

which means it's a mishap on a 2-3 (3/36 chance), fail on a 4 (also 3/36), leaving 5 a marginal (4/36), 6 (5/36) moderate, and 7+ (21/36) exceptional success.

MT also defines Impossible as 19+... given that there's effectively a DM+4 for taking extra time (but it's expressed as a difficulty shift), and given the DM+1 from stat, and DM+1 from skill, Joe Normal simply can not perform an impossible task. His max roll of 12, +4 for extra time, +1 for stat, +1 for skill is a mere 16. Even with peak stat, for DM+3 instead of +1, without a skill better than 1, Impossible remains just that: you can't get there.

A natural 2 is always a failure.

The overall scale, and Joe Normal success rates (Adj TN, chance of Adj TN, and % chance):
Simple 3+ (3+ 35/36 97.3%)
Routine 7+ (5+ 30/36 83.3%, ExTime 3+ 35/36 97.3%)
Difficult 11+ (9+ 10/36 27.7%, ExTime 5+ 30/36 83%)
Formidable 15+ (13+ 0/36 0%, ExTime 9+ 10/36 27.7%)
Impossible 19+ (17+ 0/36 0%, ExTime 13+ 0/36 0%)

The labels and processes are the same, but the definitions of Joe normal differ. Joe has a DM+2 from attribute, DM+1 from skill, and the die-roll is 1d10, not 2d6... so the odds look like:

Simple 3+ (2+ 90%) note minimum failure of 10%
Routine 7+ (4+ 70%, ExTime 2+ 90%)
Difficult 11+ (8+ 30%, ExTime 4+ 70%)
Formidable 15+ (12+ 0%, ExTime 8+ 20%)
Impossible 19+ (16+ 0%, ExTime 12+ 0%)

Twilight 2000 2E, Dark Consipiracy, and TNE

1d20 for Stat+skill or less for a difficult task. Average stat is 6, baseline skill is 1. Same labels, extra time rules, and many other similarities with MT and 2300AD... including publisher and design staff. Each level is a double or half, extra time is a full level.

For Joe Normal:
Simple x4 (19- 95%)
Routine x2 (14- 70%)
Difficult x1 (7- 50%)
Formidable x1/2 (3- 15%)
Impossible x1/4 (1- 5%)

WFRP 2E, Dark Heresy, Rogue Trader, and Deathwatch

The default task level, called challenging, for a skilled charcter is a percentile equal to attribute; this means an default level roll is roughly 31%, tho' the overall range is 17% to 70% for humans... But Routine is defined as +20 to the percentage. Trivial is +60, Difficult is -10, and Hellish is -60. Which means a hellish task is bloody hard for even the best characters, with 70% stat, +20 from skill, and +10 from a relevant trait; with +10 from help, peak chance is 50% for maxed out experience uberman.

D20 and 3.X D&D

I'm not positive I've got the labels right, but given the DCs, Joe Normal, level 1, with a +4 total bonus:

Automatic: DC5 (100% or take 10 for 100%)
Easy: DC10 (75% or take 10 for 100%)
Moderate: DC15 (50%)
Difficult: DC20 (25%)
Very Difficult: DC30 (0%)

Note that the take 10 mechanic means that Easy tasks are done automatically if skilled in them.

New World of Darkness

In World of Darkness, an average person is defined as having two "dots" in each attribute, and cursory training in a skill is one dot. Three dots gives the character three ten-sided dice. If any die comes up 8, 9, or 10, that's success, so the chance of failure is (7/10)^3, and the chance of success is 66%.

Difficult tasks may require more than one success. Each roll that comes up "10" earns another roll, so with luck a dice pool of three can attain four or more successes:

• Two or more successes: 26%
• Three or more: 6.7%
• Four or more: 1.3%
• Five or more: 0.2%

Savage Worlds

Savage Worlds uses a single die roll of a size determined by your skill level; a character of basic competence will have a d6. Roll the "target number" or higher to succeed. The default target number is four, so that's a 50-50 chance: 1, 2, 3 is failure; 4, 5, or 6 is success.

However, "wild cards," meaning PCs and important NPCs, get to roll a "wild die" too, and take the higher result. This wild die is always a d6. The higher of 2d6 against the a target number of 4 gives a 75% chance of success: failure requires both the skill die to come up 1, 2, or 3 and the wild die to do the same. Each of these has a 50% chance to happen, so the probability of both occurring is 25%.

Difficult tasks will have a higher target number, but both die rolls are "open-ended," meaning that rolling the highest possible result (6, in this case), known as an "ace," lets you roll again and add the new result, repeating as long as you keep acing.

• d6+wild die vs TN6: 30.6%
• d6+wild die vs TN8: 25.8%
• d6+wild die vs TN10: 16.0%

Burning Wheel

Burning Wheel doesn't really define "average," but a skill of B3 is common for a junior non-specialist. As with World of Darkness, high rolls count as successes, and the difficulty of a test, known as "obstacle" is defined by the number of successes required. Burning Wheel is more generous in that it uses d6s, and a 4, 5, or 6 counts as success for normal character, so each die is basically a coin flip.

• B3 vs Ob1: 87.5%
• B3 vs Ob2: 50.0%
• B3 vs Ob3: 12.5%

Generally, most systems take a baseline skill level (and if relevant, attribute level), and define task labels relative to this. They don't really give hard and fast ideas, but generally put the chances of success for a trained character at a routine task about 50-75%.

All the other breakpoints vary widely, but in general, most will have a set minimum success chance, and most also have a maximum success chance.

A few games instead label difficulties by how good you have to be to have a given chance.

The label order isn't even terribly consistent, other than the basic 5: Automatic, simple, easy, moderate, difficult. Exactly what those mean, however, varies too widely to generalize.

It truly means, no matter what system, you need to work out the percentages yourself, and you need to figure out what a particular difficulty label truly means. It also helps to figure out what the defined baseline is for those labels.

Note the extra time listings for MegaTraveller; when not rushed, a competent character can take extra time and make routine tasks fail only 2.7% of the time; even semi-skilled characters (skill level 0 but not unskilled) doing so is still that 97.3% chance of success.

Note the similar break points despite the different die-roll for 2300AD and MT... same company, similar meanings. Actually, the task system for MT was invented for use with the prior Traveller edition, by a 3rd party (Digest Group); It was then adopted and adapted for 2300AD, THEN reverted back for MegaTraveller, since MT was essentially collated by Digest Group. T2K2.2 and Dark Conspiracy have similar breakpoints... but much wider difference between peak characters and poor.

It's all just guidelines. Shortcuts to get you up to a better level of GMing competence faster, by giving you labels. But those labels have to mesh well with the skill levels listed, too. When they don't, when the odds are too out there for players, they often reject the whole system because the labels don't meet their expectations. With MT, I've lost a few players because they thought Routine was too hard as written, and no level should be labeled "Impossible"...

The Usual Approach

The usual approach is this: The labels usually work for the "Normal Guy" with "A reasonable level of skill" doing a "Normal task"...

You just need to approach all task labels from the idea of "How hard would Joe Normal find the action" and then pick the difficulty.

As a designer, the folks at GDW followed this same approach.

The Alternate Approach

A few games define difficulties by skill level at which success is expected. Fudge, as well as Fate and other such derivatives of Fudge, defines skill levels and difficulties with the same "ladder" of skill. Just define what skill level you would expect to succeed 60% of the time, pick it, and there's the difficulty.

• This is a thoughtful and good answer (already +1), but I added the bounty to see if I could get the thoughts of a few more folks here. Nov 7, 2010 at 7:23
• I'm missing something. Above @aramis writes: "Moderate: DC15 (50%)". Could someone explain how rolling 15 out of 20 is a 50% chance? I don't know DnD3.5. Is it that an average level 1 character gets +5 to rolls? Nov 9, 2010 at 18:50
• DC is rolled on Skill Total +1d20. Or, you can beat it outside combat/stress by skill total +10 (but no roll). A starting Lv 1 character with a +1 from stat and a +4 from a max skill makes that on a natural 10+ on the d20, or 55% of the time, and can automatically make it outside of combat or stress. Most starting characters can have a +5 in a skill if they choose. (Actually, given an 18 attribute, skill focus, and 4 levels of skill, up to a 10 point skill total is possible before racial bonuses.) Nov 9, 2010 at 22:56
• +50 Bounty awarded. Thanks for taking the time to answer this is such detail. Nov 12, 2010 at 18:32

In your example, XX seems to be around 50% for moderate difficulty skills. ZZZZ depends on the system. The more dice you roll and add together, the closer it approximates a Gaussian bell-curve. With just one die (such as most rolls in D20), ZZZZ is a uniform distribution. Modifiers in D20 go in increments of 1 (or 5%) for harder or easier tasks.

One aspect of RPG's that's never really sit well with my mathematical side is that with enough positive modifiers, it can be impossible to fail. With enough negative modifiers, it can be impossible to succeed. Rules like "auto success on a natural '20'" or "auto failure on a natural '1'" always seemed like kludges to me.

• @E L what about exploding dice? i.e. on a Natural 20 roll again and add, on a Natural 1 roll again and subtract? Feb 4, 2018 at 12:35