From: alongley@cape.UWaterloo.ca (Allan Longley)
Subject: Testing for Biased Dice
Originator: alongley@cape.UWaterloo.ca
Organization: University of Waterloo
Date: Tue, 10 Nov 1992 05:37:09 GMT


Good day (evening) fellow netters.

Some time ago I was talking with somebody -- now that's descriptive -- about 
testing dice for bias.  Well, here is a test to use.  I haven't actually 
tested this yet, but it should -- in theory -- work.  And no, this is not
a copy out of the old Dragon magazine, but it is the same test -- its a
pretty standard test.  I will use simple terms -- so all you math/stat
people out there, don't correct the fine points, I know.

To test a die with X sides (ie., X=4,6,8,10,12,20), roll the die 20*X times
(eg., 80 times for a d4, 400 times for a d20).  Record the number of times
each face occurs (eg., a 3 occurred 7 times) in a form similar to the first
two columns of Table 1.

	Table 1.  Example Results for a 4-sided Die
	----------------------------------------------
	value    occurrances    subtract 20    square
	----------------------------------------------
	  1          16             -4          16
	  2          24              4          16
          3          18             -2           4
          4          22              2           4
	----------------------------------------------
	                                Total = 40
	                    Indicator = 40/20 = 2.00
	----------------------------------------------

Next, subtract 20 from the number of occurances record in the second column
(ie., third column of Table 1).  Square these values (ie., square the volue
in the third column and record them in the fourth column of Table 1).  Sum 
the values in the fourth colunm (ie., Total=40) and divide this value 
by 20 to yield the indicator value (ie., Indicator=2.00).  Compare your
indicator value to the values in the Table 2 based on how many sides the die
you're testing has -- the larger the value of the Indicator the worse the
die probably is.

	Table 2.  Critical Values of the Indicator
	--------------------------------------------------------------
	Sides		Probably	Maybe		Probably
	of Die		Fair		    		Biased
	--------------------------------------------------------------
	  4		0.35		2.37		7.82
	  6		1.15		4.35		11.07 
	  8		2.17		6.35		14.07
	 10		3.33		8.34		16.92
	 12		4.58		10.34		19.68
	 20		10.12		18.34		30.14
	--------------------------------------------------------------

If you get a Indicator value below the value in the "Probably Fair" column 
then the die is probably an unbiased die.  If the value of the Indicator is 
above the value in the "Probably Biased" column then the die is probably a 
biased die.  Values between these two columns are a problem.  The column
titled "Maybe" are the Indicator values where there is a 50% chance
that the die is fair and a 50% chance that the die in biased.  If you have
gotten a Indicator value between the "Fair" and "Biased" columns then do the
test again.  If both the tests produce an Indicator value less than the
"Maybe" value then the die is probably fair -- but don't bet your life on it.
If both the tests yield an Indicator value greater than the "Maybe" column
then the die could be biased.

From Table 1, it appears that the d4 tested may be "Fair" but another test
should be done.

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Allan Longley, University of Waterloo, Department of Chemical Engineering
e-mail: alongley@cape.uwaterloo.ca
voice: (519) 885-1211			Don't make it a complex problem
home: (519) 746-1498
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           "Ph-nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn"              
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