Re: (TFT) An Even More Radical Re-Imagining of TFT...

[dwtulloh@zianet.com]

Ty Beard writes: 

> From: "Neil Gilmore" <raito@raito.com> 
>
> > > 5. A defending figure adds 5 (or 3 if using 3d6) to his roll, 
> > > but never hits a foe. 
> >
> > This definitely messes up the bell curve.
>
> Howso? If you use 3d6, the bell curve is preserved and 3 is a 
> close approximation of the effect of adding a 4th die. Is the 
> 4d6 bell curve really all that fabulous?

Ummmmm ... gee, where to begin? 

First, the 3d6 and 4d6 approximate a normal distribution, but
the fit isn't all that great in either case.  A 3d6 curve has
a mean of 10.5 and a standard deviation of 2.95804; a 4d6 has
a mean of 14.0 and a standard deviation of 3.41565.  Comparing
these to a standard normal yields the following: 

     s.d range       3d6         4d6         normal
    ==========      ======      ======       ======
   -0.5 to 0.5      0.4815      0.5216       0.3830
    0.5 to 1.5      0.2130      0.1852       0.2417
    1.5 to 2.5      0.0185      0.0532       0.0606
    2.5 to 3.5      0.0000      0.0008       0.0060 

The values for the 3d6 and 4d6 curves were computed by finding
the dice totals that most closely correspond to the s.d. range. 

    s.d value        3d6           4d6
   ==========       ======        ======
      -0.5        9.02 =>  9    12.29 => 12
       0.5       11.98 => 12    15.71 => 16
       1.5       14.94 => 15    19.12 => 19
       2.5       17.90 => 18    22.54 => 23
       3.5       20.85 => xx    25.95 => 26 

So then to determine the probability that your dice roll on
3d6 falls between 0.5 and 1.5 standard deviations from the
mean, you simply add up the probabilities associated with a
total of 13, 14 and 15.  (which equals 0.2130) 

Second, adding a value to the total of 3d6 does NOT approximate
a 4d6 curve.  All you are doing is changing the mean of the
curve, you aren't affecting its range or shape.  The 4d6 curve
has a different range and shape as compared to the 3d6 curve,
adding a number isnt going to get you there. 

Unfortunately, the 3d6 and 4d6 curves are about as close as
you are going to be able to get to the normal curve using d6.
There are other methods using dice which closer approximate a
normal curve, but you have to be willing to use other polyhedra
or re-interpret the numbers you get from a d6 throw.  ( For
example, you can get a pretty good fit for standard deviations
from the mean by throwing 3d6, counting the number of 6's and
subtracting the number of 1's.  So a roll of 6,6,1 would yield
a +1 ) 

Dan 
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