Message no. 1

From:	shadowrn@*********.com (George S Waksman)
Subject:	Shadowrun Probability
Date:	Thu Jan 10 01:00:01 2002

I notice that the question of probabilities for various rolls in Shadowrun has arisen so I
will put forth an equation I developed a few years back for calculating probabilities.

first we define the following variables

t = target number
s = threshold (or desired) number of successes
d = number of dice to be rolled

where |r| = floor of r or truncated integer

then for a single roll the chance of success is

P = (7-t+6*|t/6|-|(6*|t/6|)/t|) / (6^(|t/6|+1))

then the overall probability is

Sum[i=s to d] of (d!/(i!*(d-i)!)) * P^i * (1-P)^(d-i)


this works well for Shadowrun with a D6 system or if you want to use any other system of
dice (D8, D12, etc.) figuring the lowest number on a die is 1

then define the dollowing variable and change P as follows

x = number of sides on a die

P = (x+1-x*|t/x|-|(6*|t/x|)/t|) / (x^(|t/x|+1))

here are some tabular calculations (to 6 place) for various target numbers with 1 die
looking for 1 success just in case anyone cares

Target D4 D6 D8 D10 D12 D20
2 .750000 .833333 .875000 .900000 .916667 .950000
3 .500000 .666667 .750000 .800000 .833333 .900000
4 .250000 .500000 .625000 .700000 .750000 .850000
5 .250000 .333333 .500000 .600000 .666667 .800000
6 .187500 .166667 .375000 .500000 .583333 .750000
7 .125000 .166667 .250000 .400000 .500000 .700000
8 .062500 .138889 .125000 .300000 .416667 .650000
9 .062500 .111111 .125000 .200000 .333333 .600000
10 .046875 .083333 .109375 .100000 .250000 .550000
11 .031250 .055556 .093750 .100000 .166667 .500000
12 .015625 .027778 .078125 .090000 .083333 .450000
13 .015625 .027778 .062500 .080000 .083333 .400000
14 .011719 .023148 .046875 .070000 .076389 .350000
15 .007813 .018519 .031250 .060000 .069444 .300000
16 .003906 .138889 .015625 .050000 .062500 .250000
17 .003906 .009259 .015625 .040000 .055556 .200000
18 .002930 .004630 .013672 .030000 .048611 .150000
19 .001953 .004630 .011719 .020000 .041667 .100000
20 .000977 .003858 .009766 .010000 .034722 .050000
21 .000977 .003086 .007813 .010000 .027778 .050000
22 .000732 .002315 .005860 .009000 .020833 .047500
23 .000488 .001543 .003906 .008000 .013889 .045000
24 .000244 .000772 .001953 .007000 .006944 .042500


from this table one can conclude a few things, firstly, high target numbers are hard to
roll (GMs should note this when their player manages to roll that 24 for a monowhip or
such) and also, the dice system matters a lot, this is magnified with more dice.

-George Waksman