The following was sent to the mail list by Jay, I have stolen it for others to have a look.

The table below shows:

CDFs: cumulative mass functions (discrete distributions don't have densities--forgive my little pedantries :).

PMFs: probability mass functions.

E(R): expected value of a roll given the goal number in the left-most column.

I assumed: (a) you always get at least 1 VP and (b) a botch is worth -1.

Just so you know.

Roll CMF PMF E(R)
  d20 2d10 3d6 d20 2d10 3d6 d20 2d10 3d6
1 0.050 0 0 0.050 0 0 0 0 0
2 0.100 0.010 0 0.050 0.010 0 0.10 0 0
3 0.150 0.030 0.005 0.050 0.020 0.005 0.15 0.03 0
4 0.200 0.060 0.014 0.050 0.030 0.009 0.20 0.06 0.014
5 0.250 0.100 0.028 0.050 0.040 0.014 0.25 0.10 0.028
6 0.300 0.150 0.065 0.050 0.050 0.037 0.40 0.22 0.111
7 0.350 0.210 0.125 0.050 0.060 0.060 0.50 0.34 0.232
8 0.400 0.280 0.222 0.050 0.070 0.097 0.60 0.48 0.426
9 0.450 0.360 0.347 0.050 0.080 0.125 0.80 0.74 0.81
10 0.500 0.450 0.500 0.050 0.090 0.153 0.95 1.01 1.269
11 0.550 0.550 0.653 0.050 0.100 0.153 1.1 1.31 1.727
12 0.600 0.640 0.778 0.050 0.090 0.125 1.35 1.69 2.236
13 0.650 0.720 0.875 0.050 0.080 0.097 1.55 2.01 2.625
14 0.700 0.790 0.935 0.050 0.070 0.060 1.75 2.29 2.866
15 0.750 0.850 0.972 0.050 0.060 0.037 2.05 2.61 3.060
16 0.800 0.900 0.986 0.050 0.050 0.014 2.30 2.86 3.130
17 0.850 0.940 0.995 0.050 0.040 0.009 2.55 3.06 3.176
18 0.900 0.970 1.00 0.050 0.030 0.005 2.9 3.26 3.185
19 0.950 0.990 1.00 0.050 0.020 0.000 2.9 3.26 3.185
20 1.00 1.00 1.00 0.050 0.010 0.000 2.9 3.26 3.185

Jay Verkuilen jayv@uiuc.edu