Wednesday 4 February 2015

Non-transitive dice

A set of 3 non-transitive dice can be bought from Maths Gear. They have sides like this:

Die A: 333336
Die B: 222555
Die C: 144444   $P(A>B)=P(B>C)=21/36$ and $P(C>A)=25/36$.

These are optimal, in that they maximize the smallest winning probability. However, there are other dice for which the sum of the 3 winning probabilities is greater.

Die A: 114444
Die B: 333333
Die C: 222255   $P(A>B)=P(B>C)=24/36$ and $P(C>A)=20/36$. 

There is an interesting article by James Grime (of the Millennium Mathematics Project) in which he says lots more about non-transitive dice. He has devised a set of 5 non-transitive dice. In another interesting article there is a video in which James discusses non-transitive dice with David Spiegelhalter.

James made a contribution to this blog last year, and you can read his comment at the end of the post here: http://weberprobability.blogspot.co.uk/2014/02/nontransitive-dice.html