Running the numbers

[sonarean]

So, a few posts ago, I talked about running the numbers on dice pool vs single die systems.  My premise was that you're far less likely to fail in a dice pool system than with a single die + modifiers roll.  Naturally, it depends on the size of the pool, but here are my findings for both d6 (a la Shadowrun) and d10 (a la nWoD)  systems:  (Note, I'm only interested in a guaranteed failure (no successes), not degrees of success; the equivalent to rolling a natural 1 on a d20)

For 1d6, the possible rolls follow (successes highlighted)
1    2    3    4    5    6

So the odds of a success are 2/6 or approximately 33%

For 2d6, the possible rolls follow (again, successes highlighted)

1,1    1,2    1,3    1,4    1,5    1,6
2,1    2,2    2,3    2,4    2,5    2,6
3,1    3,2    3,3    3,4    3,5    3,6
4,1    4,2    4,3    4,4    4,5    4,6     This can be expressed as 36-16
5,1    5,2    5,3     5,4    5,5    5,6                                                    36
6,1    6,2    6,3     6,4    6,5    6,6

So the odds of a success are 20/36 or approximately 55%

I'll not bore you with the permutations for higher numbers of dice and jump straight to the results.  The generic equation for the success rate can be expressed as follows:  (I apologize for LJ not being the best medium for math discussions)

(number of sides)^(number of dice) - (number of sides that fail)^(number of dice)
                                 (number of sides)^(number of dice)

Using this equation, the following chart can be generated:  (note these are approximations)

# of diced6 success rated10 success rate10.3333330.320.5555560.5130.7037040.65740.8024690.759950.8683130.8319360.9122090.88235170.9414720.91764680.9609820.94235290.9739880.959646100.9826580.971752
So, assuming all things being equal and that even a 2 on a d20 has a chance of success, you'd need a pool of 8d6 or 9d10 to get up to the 95% success rate of a d20.  Now, since many d20 systems are scaled to an approximately 50% success rate (see Morrison's 4e math breakdown morrisonmp.wordpress.com/2010/03/08/doing-the-math/#more-48) we can see that with a pool of only 3 dice we achieve a much higher success rate than a single d20.

Note:  These numbers do not account for "glitches" in the Shadowrun system.  I might do something with those later.  We'll see.

tl;dr version:
Dice pool systems grant a much higher chance of success than a single die system.  Thanks to Scott for keeping me entertained for an evening.