Graphs of Success Probability by Skill Total and Difficulty

I've given you tables of success probabilities by skill total and difficulty for two systems (World of Darkness, Shadowrun 4th ed.), plus a graph for Heavy Gear.  Here I present that information again in graphs, plus two more systems, to show some of the different patterns that exist for success probabilities with increases in skill among different systems.

Linear

Here is your standard d20 system, most popular in Dungeons and Dragons.  Each character has a skill modified by an attribute and various other junk, added to a d20 result and compared to a difficulty level.  Each increase in the skill total raises the probability of success by 5% linearly.  There is always at least a 5% chance of failure (rolling a 1).  In the D&D games, skills are not bought with general character development points, but characters are alloted a few points each level to be used only for skills.  Difficulty levels typically scale with character levels, so it behooves players to specialize in a few skills that are always increased with the character level in order to maintain good probabilities of success as characters level up.  I am not getting in to "taking 10" or "taking 20".

Inconsistent

Here is the graph for Dream Pod 9's Silhouette system, used in their Heavy Gear game.  We can see that the progression is not consistent.  The lowest skill is concave, rapidly dropping the probability of success at low difficulties relative to the drop at higher difficulties where the probability of success is already very low.  A skill of 1 has a linear descent.  Higher skills progressively maintain high success rates among lower difficulties before rapidly plunging at higher difficulties, and then there is the bent tail as it becomes more possible to roll multiple 6s.  Attribute bonuses are added to skill roll results, shifting the graph to the right without changing its shape.

Normal

Isn't that pretty?  I am not sure if I am completely representing the GURPS system accurately here, but I think players just have to roll lower than the characters' skills on 3d6 to succeed at tasks (17s and 18s fail).  So, there is no real "difficulty level" for tasks other than what is forced by skill levels.  There may be modifiers that increase or decrease a skill for the purpose of a challenge, shifting the whole curve to the left or right.  If we graphed the probabilities of each individual outcome for 3d6, the line would be shaped like a bell.  I call this "normal" because as a "normal distribution" it has higher probabilities of outcomes in the middle, progressively less likely outcomes away from the middle, and is relatively symmetrical.

Inconsistent Normal

We can see here that both Shadowrun by Catalyst Game Labs and World of Darkness by White Wolf approach the normal curve as their dice pools (skill total, or skill + attribute) increase.  With few dice in these systems, it is impossible to approximate the distribution of the normal pattern, and the results more follow the Inconsistent pattern.  These systems both involve rolling multiple dice (d6 and d10, respectively), and counting die results over a threshold as "successes".  Players need a number of successes equal to a task's difficult in order to succeed.  So, the terminology can get annoying as people get a bunch of successes but still fail at a task.

I really like how the Normal distribution of probabilities of success works in simulations, but not necessarily the way that GURPS implements it in the absence of difficulty levels.  In real life, when we encounter tasks far below our skill level, we are quite likely to succeed at them and have a low variance with our high success rate.  When we encounter tasks far above our skill level, we are quite likely to fail at them and have a low variance with our high failure rate.  Tasks closer to our skill level have increasingly variant success rates.  Because of this, I am in favor of the use of normal distributions of probability of success in simulation systems.  This typically requires rolling more than one die and summing the results.

White Wolf's World of Darkness Probability Tables

White Wolf's game mechanics typically look as though they were created by coked-up baboons.  I experienced a lot of frustration creating the probability tables for their 2004 version of the World of Darkness because of the mechanic of re-rolling 10s.  The whole game is like rolling damage for the arquebus in old D&D.  Technically, it is possible to roll forever, so I set some limits.  I stop calculating when the probability of an outcome drops below 0.001, or after the third roll.  It was relatively easy to find the probabilities of up to three successes with one die, then much more complicated for two dice, then after some struggles I noticed a pattern in how many separate equations must be solved and combined for each possible outcome (that meets my criteria) for a given number of dice:

So, instead of trying to solve over two thousand equations or trying to figure out how to write a computer program that would do it for me, I brute forced the problem.  I used Excel to whip up 10,000 sets of 3 rolls, then mixed them up twenty different ways, then averaged the occurrences of each number of successes.  The values in the following tables should be correct to about +/- 0.002.  (Click on the pictures to see bigger versions. I'm annoyed with blogspot's layouts.)

And, of course, by adding from the right we can find the probabilities of rolling at least any specific number of successes.  Remember that these probabilities include up to two re-rolls of 10s.

I hope that these tables will be useful to game masters (storytellers) in determining difficulty levels of challenges for characters in their campaigns.  They may also be helpful to players in deciding how much to increase a skill or attribute.