A set of nontransitive dice is a set of dice for which the relation "is more likely to roll a higher number" is not transitive. See also intransitivity.

This situation is similar to that in the game Rock, Paper, Scissors, in which each element has an advantage over one choice and a disadvantage to the other.Consider a set of three dice, A, B and C such that

• die A has sides {2,2,4,4,9,9},
• die B has sides {1,1,6,6,8,8}, and
• die C has sides {3,3,5,5,7,7}.

Then:

• the probability that A rolls a higher number than B is 5/9 (55.55 %),
• the probability that B rolls a higher number than C is 5/9, and
• the probability that C rolls a higher number than A is 5/9.

Thus A is more likely to roll a higher number than B, B is more likely to roll a higher number than C, and C is more likely to roll a higher number than A. This shows that the relation "is more likely to roll a higher number" is not transitive with these dice, and so we say this is a set of nontransitive dice.


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