From: alongley@cape.UWaterloo.ca (Allan Longley)
Subject: Re: Testing for Biased Dice
Organization: University of Waterloo
Date: Tue, 10 Nov 1992 16:44:16 GMT


dcgonyie@eos.ncsu.edu writes:

> There are a few dice I have that need to be tested, I think...
> The only thing I don't like about this system is having to roll that one
> die 100's of times...

Well, you could roll the die fewer number of times, but your chance of
getting a meaningful result decreases.  The test could be changed to using
half the number of rolls as follows:

To test a die with X sides (ie., X=4,6,8,10,12,20), roll the die 10*X times
(eg., 40 times for a d4, 200 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 10    square
	----------------------------------------------
	  1           8             -2           4
	  2          12              2           4
          3           9             -1           1
          4          11              1           1
	----------------------------------------------
	                                Total = 10
	                    Indicator = 10/10 = 1.00
	----------------------------------------------

Next, subtract 10 from the number of occurances record in the second column
(ie., third column of Table 1).  Square these values (ie., square the value
in the third column and record them in the fourth column of Table 1).  Sum 
the values in the fourth colunm (ie., Total=10) and divide this value 
by 10 to yield the indicator value (ie., Indicator=1.00).  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.

-----------------------------------------------------------------------------
Allan Longley, University of Waterloo, Department of Chemical Engineering
e-mail: alongley@cape.uwaterloo.ca
voice: (519) 885-1211			Stats can NEVER prove something
home: (519) 746-1498			is true!
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