Non-Transitive Dice Are Weird

Non-transitive dice are beyond weird. As seen in the image below of Bradley Efron’s four-dice nontransitive set, these dice are not like normal six-sided dice in that each has a unique pattern of numbers.

Imagine setting up a game where each person picks a die and then rolls that die 50 times. Whichever die rolls higher, the most times wins.

The purple die will roll a higher number than the yellow die 2/3rds of the time.

The yellow die, in turn, will roll a higher number than the red die 2/3rds of the time.

Finally, the red die will roll a higher number than the green die 2/3rds of the time.

Here’s where the weirdness of the non-transitive nature of the dice comes into play. The green die will roll a higher number than the purple die 2/3rds of the time.

Wikipedia has a nice entry on non-transitive dice, including a detailed look at the probabilities of the Efron dice. The set of Efron’s dice in the image is available for $5 plus shipping from the National Museum of Mathematics.

Non-Transitive Dice
Non-Transitive Dice