# (TFT) Crunching the numbers...

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I found Dan Tulloh's suggestion about taking the
highest three of X dice to be quite intriguing.
The math is beyond me, too, so I wrote up a die-
rolling program and cranked out 50,000 6d6, 5d6,
and 4d6 rolls, taking the highest three results,
and noting how many times each possible result
came up.  From there it was easy to get the
approximate percentages of each result, and
then the chance of rolling any given number
or less, which is basically the chance of
success.  Here's what I came up with, along
with the (actual, not approximate) percentages
that exist with a normal 3d6 roll:

Chance of success, taking the highest three
dice, if required number for success (adjDX
or whatever) is between 3 and 18:
6d6        5d6        4d6        3d6
3     0.06       0.28       1.12       4.63
4     0.06       0.28       1.12       4.63
5     0.06       0.28       1.12       4.63
6     0.20       0.80       2.67       9.26
7     0.61       2.01       5.55      16.20
8     1.54       4.20      10.14      25.92
9     3.45       8.05      17.06      37.49
10    6.95      14.07      26.48      49.99
11   12.81      22.66      38.06      62.49
12   21.72      33.95      51.00      74.06
13   33.78      47.58      64.41      83.78
14   48.86      62.32      77.04      90.72
15   65.68      76.64      87.09      95.35
16   65.68      76.64      87.09      95.35
17   65.68      76.64      87.09      95.35
18   65.68      76.64      87.09      95.35

Compare this with a similar success chart
for 4+ die rolls given TFT's usual method:

Chance of success, summation method
6d6        5d6        4d6        3d6
3:    6.08       5.87       5.40       0.46
4:    6.08       5.87       5.40       1.85
5:    6.08       5.87       5.40       4.63*
6:    6.08       5.87       5.40       9.26
7:    6.08       5.87       5.40      16.20
8:    6.08       5.87       5.40*     25.92
9:    6.08       5.87       9.72      37.49
10:   6.08       5.87      15.89      49.99
11:   6.08       5.87*     23.91      62.49
12:   6.08       9.79      33.56      74.06
13:   6.08       15.19     44.36      83.78
14:   6.08*      22.13     55.63      90.72
15:   9.65       30.50     66.43      95.35
16:  14.47       39.95     76.08      95.35
17:  20.59       49.98     84.10      95.35
18:  27.94       60.01     90.27      95.35

*Threshold of automatic success (ITL, p.38)

Obviously rolls much higher than 18 are
possible when summing 4+ dice, but those
figures aren't needed for comparison with
the Highest Three method.)

A couple things of interest.  Note that the
Highest Three method produces a much "smoother"
progression of difficulty for rolls vs. moderate
stat levels (say, 12-14).  People with stats in
this range just have no business trying to make
6d6 rolls under TFT's summation system, but they
have a reasonable shot at it with the Highest
Three system.  Conversely, having a stat higher
than 15 is of less importance with the Highest
Three system, because a roll of 16+ is automatic
failure whether you have a 15 or a 25 in the
stat--and as you can see the chance of rolling
a 16+ becomes quite great when you're taking
the highest three of 4 or more dice.

It seems like the original TFT method is designed
with mega-level stats in mind, while the Highest
Three system works better for people with "normal"
stats...say, 18 or less.  Under Highest Three it
doesn't really matter if your DX is 16 or 25, your
chance of making a 6d6 DX roll is the same.
Under the original summation system that 25 DX
will help at making such rolls.

Of course, how many campaigns really have people
with 25 DX?  It seems to me that it might be better
to suit the mechanics to the level of play that is
more prevalent, where people have moderate stats,
and that's exactly what the Highest Three
system does.

There is one aspect of the Highest Three system
that troubles me, though, and that's critical
success/failure.  The TFT system maintains a
roughly 2% chance of critical success (a 3 or
4, when rolling 3 dice), regardless of the number
of dice being rolled.
But obviously when you're selecting the highest
three of five or six dice, you're going to get a
lot of 17's and 18's, and not many 3's and 4's:
6d6,     5d6     4d6
Crit Failure:    18%     12%     6%
Crit Success:   0.01%   0.08%   0.42%

These numbers are based on the same 50k rolls
that I did for the first table.

While one can argue that the chance of critical failure
*should* be higher when you're trying a 5d6 roll, I'm
not sure that it should be six times higher.  And
critical success...well, it's one in ten-thousand with
a 6d6 roll.  One possible way around this would be to
always roll three red dice, regardless of how many dice
are being used to make a roll.  That way you could
always base the occurrence of critical results on the
red dice, while using the highest three to determine
success/failure.  Or is that gettting too complicated?

Interesting stuff...
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