The Mirrored Road into the Interior

Today has been math day for Michael. I've got a decent intuitive grasp of mathematics, but there are some fairly serious holes in my knowledge base. It means that I often have to figure things out from earlier principles. Today's "taking more time than it should have" issue has been dice math.
Particularly I wanted to take a look at the math behind RPG systems that use a success dice sort of system. When you roll dice to see if an action is successful, there are several ways to handle it. Many games have you roll a die or several dice (1d20 for D&D, 2d6 for Big Eyes Small Mouth, 1d100 for the old call of Cthullhu systems, 2d10 or 3d6 for the Fuzion based games, etc) add assorted modifiers and then compare your total to a target number. If your total is equal or higher than the number, you succeed. Otherwise, you fail. There are some important differences between the systems that role a single die and the ones that roll multiple dice, but the comparison methods are the same. Critical hits and misses make more sense the multiple dice systems, but your rolls tend to be clustered at the center of the range a lot more. (There are other dice methods that work on the comparison to a target number, but these are the two major categories.)
 personally prefer the straight probabilities you get from a single die, but systems with multiple dice do have some interesting features.

Target Value	Chance of success
4	99.4
5	98.1
6	95.4
7	90.7
8	83.8
9	74.1
10	62.5
11	50
12	37.5
13	25.9
14	16.2
15	9.3
16	4.6
17	1.9
18	0.5

In the above table, notice that there are some reasonable probability spreads. If I think something should be 30% likely to work for a normal person, I set the target score to 13 or so. It isn't perfect, but there is a reasonable value for any 10% step between 0 and 100%.On the other hand, bonuses can quickly skew the odds. A +2 bonus on the above role moves the chance from 1 in 4 to 1 in 2. Given a 30% chance in a d20 setting, a +2 bonus shifts you to a 40 percent change. +2 is always a 10 percentage point shift when you are using a d20.

The other sort of system is a success dice system. Here you have a pool of dice that varies based on your statistics and skills. To do something, you roll your pool of dice, count all of the of the dice that ended up above the system's target number, and count successes. The number of successes determines if you actually succeeded or not. For example, you might have a dice pool of 10d10, with a target number of 8. If an action requires 4 successes, then you need to roll 4 or more 8s or better. In this case, you have about a 35% chance of actually succeeding. It is a serious pain in the ass to modify difficulty in a system like this except in a few very limited cases.

Here's the percentages for different numbers of required successes with th above 10d10 pool.

10 dice
1	97.18%
2	85.07%
3	61.72%
4	35.04%
5	15.03%
6	4.73%
7	1.06%
8	0.16%
9	0.01%
10	0.01%

(that last .01% is actually ..0059%)