This problem follows naturally
from X-Dice, although it may be attempted
A company wishes
to produce a set of three different 6-sided dice coloured Apple
Green, Bright Pink and Cool Grey, called $A, B$ and $C$
They are to be made with the following properties:
1. The faces are to be numbered using only whole numbers 1 to
2. Some of the numbers 1 to 6 can be left out or repeated as
desired on each dice.
3. Apple Green is expected to beat Bright Pink on a single
4. Bright Pink is expected to beat Cool Grey on a single
5. Cool Grey is expected to beat Apple Green on a single
Invent a set of such dice.
What is the probability that each of your dice wins or loses over
each of the other dice?
Is it possible to create a totally fair set of such dice with
$P(A> B) = P(B> C) = P(C> A)$?
NOTES AND BACKGROUND
These dice exhibit a property called non-transitivity. Other
examples of non-transitivity are found in voting systems. You can
read about this in our article Transitivity
. You might also
wish to try the related problem A Dicey Paradox