Computer Dice

 FIBS: Analysis of 10 million rolls

 From: Stephen Turner Address: sret1@cam.ac.uk Date: 5 April 1997 Subject: Re: Randomness and FIBS dice (with 10 MILLION ROLLS!!) Forum: rec.games.backgammon Google: 334636DB.6F9B@cam.ac.uk

```Executive summary: FIBS dice appear fair on the tests I carried out.
Details follow.

First some background. Patti's sample of 5,530,616 rolls showed 924,319
doubles. If the dice were fair, this many doubles or more would only come
up 1 in 530 times (0.189%), so there was some cause for concern.

Just before Robert-Jan Veldhuizen noticed this, the FIBS command 'matrix'
was also pointed out to me by someone else, and I had started collecting
data from it. Here is my original post:

---------------------------------------------------------------------------
From: Stephen Turner <sret1@statslab.cam.ac.uk>
Subject: Re: 5 million rolls
Date: Tue, 29 Oct 1996 11:31:53 +0000

I too have been carrying out some tests of the FIBS dice in the last couple
command because it's not on the list of commands, but it produces data not
only on the frequency of rolls, but on the frequency of each of the
possible 1296 possible pairs of consecutive rolls. So seeing whether these
data are fair will test both whether some rolls come up more than others,
and whether some rolls are more likely than others to come up after
specific rolls.

I won't bore you with the details of another statistical test, except to
say for the benefit of statisticians that it's a simple chi squared test on
1295 degrees of freedom, which I approximate by N(1295,2590). All everyone
else needs to know is that it comes out at the end with a number, which we
hope is near 0. A number greater than 2 indicates that the dice are biased
in some way (too many of one roll or of one pair of consecutive rolls at
the expense of another). A number less than -2 indicates that the dice are
too good -- in the sense that it looks as if they're fixed to try and get
the right proportions, and there's actually too little statistical
variation.

I shall continue to collect more rolls until I have reached 10 million.

Overall, I think we can say that the jury is still out. There may be a
deviation from unskewed-ness, but if so, it is slight, and more data are
needed to be sure. I think that it probably wouldn't affect the game,
UNLESS there is some reason why any imperfections are more likely to occur
at certain times, such as during races, doubles after doubles, etc.
---------------------------------------------------------------------------

In the end, the data took me longer to collect than I had hoped because I
didn't log on to FIBS very often. I collected 14 sets of data over nearly
6 months from 11th October to 5th April. This was a total of 10,593,121
rolls, so 10,593,107 pairs of consecutive rolls.

I can supply the data on request, but the chi-squared statistic gave an
answer of -0.355521, showing no evidence of bias. As I said above, this
test will check both whether some rolls come up more than others, and
whether some rolls are more likely than others to come up after specific
rolls. As a check, the number of doubles in the sample was 1763911, or
16.651%, fractionally less than expected, but well within normal
statistical variation (1.32 s.d.s from the mean, in fact).

Of course this doesn't check for every possible bias in the data, but it
checks for the most plausible and most widely suggested ones.

--
Stephen Turner    sret1@cam.ac.uk    http://www.statslab.cam.ac.uk/~sret1/
Statistical Laboratory,  16 Mill Lane, Cambridge, CB2 1SB, England
"This store will remain open during modernisation. We apologise
for any inconvenience this may cause" Topshop, Cambridge
```

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