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Geek warning: the following description is starting to get pretty geeky.
I've set follow-ups to lugnet.off-topic.geek, see you there if you want
to continue discussion...
Christopher Weeks wrote:
>
> In lugnet.fun.gaming, Frank Filz writes:
>
> > The most interesting thing about the system was
> > how it managed to use a true normal distribution without much pain at
> > all. Basically, instead of trying to roll an effect number with the
> > dice, you generated a random number between 0 and 1 (exclusive) by
> > generating the digits. A simple chart then converted the fraction to a
> > positive or negative adjustment which was added to a skill and then
> > compared to the defensive skill or a fixed number. Anyone who could
> > remember numbers at all could easily remember the first one or two
> > standard deviations (the adjustments were about 3 [or is it 6] steps per
> > standard deviation). I still carry the chart in my wallet, and I could
> > actually run a game of this without having anything else if pressed
> > (though I would have to quickly reconstruct some charts from memory).
> > The way the chart works, you basically need two digits beyond a leading
> > string of 0s or 9s. You don't tend to need to know the bottom half of
> > the chart much because usually those are misses (and I firmly believe in
> > don't waste time trying to determine if you missed by 3 or 5 points).
>
> Frank, can you explain this in more detail? I've been trying to reconcile the
> bits that you mention into a roll-resolution system and I'm not getting it.
>
> When you talk about generating digits, I'm thinking that you're rolling d10s as
> you would percentiles (except any number, not just two) or something. But
> then, there's no standard deviation on a flat roll. So I guess you're just
> comparing to some standard standard deviation numbers...thus the chart to which
> you refer.
>
> But that's not sounding right. And what is your reference to "steps" per
> standard deviation?
Ok,
First, for those who want to go into more depth of the math, you could
start with this web site:
http://www.pinkmonkey.com/studyguides/subjects/stats/chap7/s0707901.asp
Ok, so the basics: A "normal distribution" is a probability curve which
has an area under it equal to one. The curve is centered around zero
with +1 and -1 being the bounds of one standard deviation.
Generally, we are interested in the probability of achieving some
minimal result (What's the probability of hitting an opponent, given my
attack skill and his defense skill?).
A convenient way to use a random number is to generate a random number
somehow, and add it to the attackers skill, and compare against the
defenders skill. Of course if your random number is generated by rolling
standard dice, you need to add something to the defender's skill.
The normal distribution is good for working with random numbers which
are centered on 0. Of course, a standard deviation of +-1 is a little
harsh for gaming, so it is best to scale the distribution. The system I
was using scaled so approximately +-7 was one standard deviation (I
think it may be +-6.666666... but I'm not sure). It is easy to generate
a table from the normal distribution which gives the probability of
achieving any result or lower (so on the standardized normal
distribution where one standard deviation is +-1, the following table
can be generated:
result probability fraction
-3 0.14% .0014
-2 2.28% .0228
-1 15.88% .1588
0 50.00% .5000
+1 84.13% .8413
+2 97.72% .9772
+3 99.86% .9986
You can find detailed tables in various math books. What is so cool
about the system my friend developed is the simple way it derived to
generate a real normal distribution. What if rather than randomly
generating the result numbers, we randomly generate the probability?
Well, there's an easy way to generate these numbers, just roll 1d10 for
each digit starting at the decimal point. Now if you simplify the chart
a little bit, you can set up a pretty accurate chart in which you only
generate two "significant" digits, in other words, generate two digits
past a leading string of all 0s or all 9s.
So, the table used in the game looks like this (we can see that +-20 is
very close to 3 standard deviations, I've included a few numbers past
the 3 standard deviation mark, my chart is complete from -40 to +40):
-40 .00000000099
-35 .000000076
-30 .0000034
-25 .000088
-20 .0013
-19 .0022
-18 .0035
-17 .0054
-16 .0082
-15 .012
-14 .018
-13 .025
-12 .036
-11 .049
-10 .067
-9 .088
-8 .12
-7 .15
-6 .18
-5 .23
-4 .27
-3 .33
-2 .38
-1 .44
0 .50
+1 .56
+2 .62
+3 .67
+4 .73
+5 .77
+6 .82
+7 .85
+8 .88
+9 .912
+10 .933
+11 .951
+12 .964
+13 .975
+14 .982
+15 .988
+16 .9918
+17 .9946
+18 .9965
+19 .9978
+20 .9987
+25 .999912
+30 .9999966
+35 .999999924
+40 .99999999901
Now since rarely does a negative result hit (it seems generally good in
games to have hits occur less than 50% of the time between matched
opponents), and the large percentage of the time, you are rolling within
a standard deviation or two, it turns out to be possible to memorize the
table. The result is a system which doesn't take much time more than any
other resolution system, yet is a pure normal distribution. Also, the
way the table is created, it is possible to scale it to suit your needs,
though from play experience I would suggest the above chart is close to
ideal.
For comparison, here is a similar chart for 2d6:
2 0.00
3 0.06
4 0.14
5 0.25
6 0.39
7 0.56
8 0.69
9 0.81
10 0.89
11 0.94
12 0.97
Note that since 2d6 has a finite range, the first entry is 0 since every
result is at least 2. If we put 13 on the chart, it would be 1.00 (and
thus unachievable). Note that in rolling on such a table, while it feels
like you're rolling percentile dice in which 00 is 100 not 0 in most
systems (if 00 was 0, then the chance of getting a 20 or less would be
21% which would feel odd). Since we are generating digits of a fraction,
00 is 0.
FUT: lugnet.off-topic.geek
Frank
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Message has 1 Reply:
 | | Re: Elements of a brick oriented RPG
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| (...) I did some more investigating and figured out some useful stuff... (...) Excel has a useful function NORMDIST, you can generate the above table using: =NORMDIST(A1,0,1,TRUE) Excel generates slightly different values: -3 .0013 -2 .0228 -1 .1587 (...) (22 years ago, 13-Aug-02, to lugnet.off-topic.geek)
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Message is in Reply To:
| | Re: Elements of a brick oriented RPG
|
| (...) Frank, can you explain this in more detail? I've been trying to reconcile the bits that you mention into a roll-resolution system and I'm not getting it. When you talk about generating digits, I'm thinking that you're rolling d10s as you would (...) (22 years ago, 28-May-02, to lugnet.gaming)
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