Alternate Section 4.2, Rolling the Dice
=======================================

By: Reimer Behrends, r_behren@informatik.uni-kl.de

As some people dislike the recommended dice method in the basic FUDGE
rules (it tends to favour extreme results in an unrealistic way), here
are several methods to roll dice in a fashion that is more amenable to
reality checks (well, of course no method can be fully realistic, but
the following methods reduce the probability of achieving extreme
results).

1. Using d3

Some manufacturers make d3s by numbering six-sided dice from 1 to 3
twice. Rolling 2d3-2d3 (best accomplished by using dice of different
colours or instead computing 4d3-8) yields the following probabilities:

 n | P(x=n) | P(x>=n)
---------------------
-4 |   1.2% | 100.0%
-3 |   4.9% |  98.8%
-2 |  12.3% |  93.8%
-1 |  19.8% |  81.5%
 0 |  23.5% |  61.7%
+1 |  19.8% |  38.3%
+2 |  12.3% |  18.5%
+3 |   4.9% |   6.2%
+4 |   1.2% |   1.2%

2. Using custom dice

Numbering blank six-sided dice with [-2,-1,0,0,+1,+2] gives a nice
distribution, too:

 n | P(x=n) | P(x>=n)
---------------------
-4 |   2.7% | 100.0%
-3 |   5.6% |  97.2%
-2 |  13.9% |  91.7%
-1 |  16.7% |  77.8%
 0 |  22.2% |  61.1%
+1 |  16.7% |  38.9%
+2 |  13.9% |  22.2%
+3 |   5.6% |   8.3%
+4 |   2.7% |   2.7%

3. Using a d20 or d30 with a table

The ease of having only to use a single die (and therefore avoiding dice
arithmetic) for some people offsets the necessity to use a table to look
the real result up. Both d20 and d30 are suitable for this purpose.
However, the d20 method is a bit more complicated, requiring a reroll
for every 1 or 20 rolled (which is rare enough to make it unobtrusive).
For a d20, use the following table:

Rolled |  1 | 2-3 | 4-7 | 8-13 | 14-17 | 18-19 | 20
---------------------------------------------------
Result | -3 |  -2 |  -1 |    0 |    +1 |    +2 | +3

To get the full range of -4 to +4 results, reroll all 1s and 20s. If the
second roll after a 1 is less than or equal to 7 (i.e. a negative result
on the above table), count is as a -4. For a rolled 20, roll again and
count everything that is a 14 or above (i.e. a positive result on the
above table) as a +4. And if you want the full range of -6 through +6 as
possible results, just adding the second roll if it yields a result with
the same sign as the first one, you'll even have a slight possibility
for Terrible results even if the skill level is Superb (Probability is
0.25%, i.e. 1 out of 400 rolls on the average) and vice versa. But note
that a 20 followed by a 1 still is a +3 result and not a 0 (and the same
of course for other results with opposite signs)!

Probabilities for the first variant are as follows:

 n | P(x=n) | P(x>=n)
---------------------
-4 |   1.8% | 100.0%
-3 |   3.3% |  98.3%
-2 |  10.0% |  95.0%
-1 |  20.0% |  85.0%
 0 |  30.0% |  65.0%
+1 |  20.0% |  35.0%
+2 |  10.0% |  15.0%
+3 |   3.3% |   5.0%
+4 |   1.8% |   1.7%

The open-ended variant yields the following probabilities:

 n | P(x=n) | P(x>=n)
---------------------
-6 |   0.3% |  99.8%
-5 |   0.5% |  99.5%
-4 |   1.0% |  99.0%
-3 |   3.3% |  98.3%
-2 |  10.0% |  95.0%
-1 |  20.0% |  85.0%
 0 |  30.0% |  65.0%
+1 |  20.0% |  35.0%
+2 |  10.0% |  15.0%
+3 |   3.3% |   5.0%
+4 |   1.0% |   1.7%
+5 |   0.5% |   0.8%
+6 |   0.3% |   0.3%

With a d30, use the following table:

Rolled |  1 | 2-3 | 4-6 | 7-11 | 12-19 | 20-24 | 25-27 | 28-29 | 30
-------------------------------------------------------------------
Result | -4 |  -3 |  -2 |   -1 |     0 |    +1 |    +2 |    +3 | +4

This results in the following probability distribution:

 n | P(x=n) | P(x>=n)
---------------------
-4 |   3.3% | 100.0%
-3 |   6.7% |  96.7%
-2 |  10.0% |  90.0%
-1 |  16.7% |  80.0%
 0 |  26.7% |  63.3%
+1 |  16.7% |  36.7%
+2 |  10.0% |  20.0%
+3 |   6.7% |  10.0%
+4 |   3.3% |   3.3%

4. Using d12

Good results can be achieved by the formula 1d12-1d12 followed by
interpretation on the following table.

<= -10 | -9..-8 | -7..-5 | -4..-2 | -1..+1 | +2..+4 | +5..+7 | +8..+9 | >= +10
------------------------------------------------------------------------------
    -4 |     -3 |     -2 |     -1 |      0 |     +1 |     +2 |     +3 |     +4

This is a bit cumbersome, but if you multiply each numeric trait level
by three (resulting in Terrible = -9, Poor = -6, Medicore = -3, Fair =
0, Good = +3, Great = +6, Superb = +9) and then just add the dice and
round to the nearest multiple of three, this is easy to handle.  (Well,
you have to treat rolls of +/-10 differently, but if you remember that
every two-digit result is an effective +/-4, this is easy, too.  Even
more so since such a result only comes up once per 24 rolls on the
average.) In addition, you can add intermediate levels if you think the
standard levels are too granular. Probabilities are as follows:

 n | P(x=n) | P(x>=n)
---------------------
-4 |   2.1% | 100.0%
-3 |   4.9% |  97.9%
-2 |  12.5% |  93.1%
-1 |  18.8% |  80.6%
 0 |  23.6% |  61.8%
+1 |  18.8% |  38.2%
+2 |  12.5% |  19.4%
+3 |   4.9% |   6.9%
+4 |   2.1% |   2.1%