# a→ab

## computation, game design, and experimentation

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# Strange Dice

January 13, 2014
tags: dice, fudge, probabilities

So, in the Fate mailing list, people are discussing what the various effects of replacing a Fudge die with a normal polyhedral die are. This is an interesting conversation to me, and I thought I'd do some math to get a grip on it...

# 4dF

First up is the standard 4dF distribution (which I've talked about before)

4dF | %
----|-------------
-4|1.23456790123
-3|4.93827160494
-2|12.3456790123
-1|19.7530864198
0|23.4567901235
+1|19.7530864198
+2|12.3456790123
+3|4.93827160494
+4|1.23456790123


This is a baseline for my comparisons below.

# 3dF + 1d4

An interesting option is replacing 1dF by 1d4.

3dF + 1d4 | %
----------|--------------
-2|0.925925925926
-1|3.7037037037
0|9.25925925926
+1|15.7407407407
+2|20.3703703704
+3|20.3703703704
+4|15.7407407407
+5|9.25925925926
+6|3.7037037037
+7|0.925925925926


Here it's shifted the average from 0 to +2.5, and increased the range--but not uniformly! The low end has been shifted from -4 to -2, a +2 increase, while the high end has been shifted from +4 to +7, a +3 increase!

# 3dF + 1d6

A bit more extreme is replacing 1dF by 1d6.

3dF + 1d6 | %
----------|--------------
-2|0.617283950617
-1|2.46913580247
0|6.17283950617
+1|10.4938271605
+2|14.1975308642
+3|16.049382716
+4|16.049382716
+5|14.1975308642
+6|10.4938271605
+7|6.17283950617
+8|2.46913580247
+9|0.617283950617


The average has shifted from 0 to +3.5, a significant boost. Again, the low end has shifted by +2, but the high end has shifted from +4 to +9, a +5 shift! This seems very high-powered.

# 2dF + 2d4

A rather extreme option, this removes negative results from the equation altogether... though it does make a very nice bell curve centered around 5 in the range [0,10]. This could be very handy as a normalized d10 replacement, perhaps... but so could 4dF + 4, and that wouldn't have to worry about rerolling on 10. Anyways..

2dF + 2d4 | %
----------|--------------
0|0.694444444444
+1|2.77777777778
+2|6.94444444444
+3|12.5
+4|17.3611111111
+5|19.4444444444
+6|17.3611111111
+7|12.5
+8|6.94444444444
+9|2.77777777778
+10|0.694444444444


Neat to me, perhaps not so much to others... If you're interested, here's an anydice link where you can see graphs and such, and play around with it yourself.