S Dice paradox. Two dice D1 and D2 are labelled as follows.

D1 W 6 3 3 3 3 3 ; D2 W 5 5 5 2 2 2 :

Andy and Beth roll D1 and D2 respectively. Whoever gets the higher number wins.

(a) Show that Andy has a higher probability of winning; we write this as D1  D2.

(b) Beth notices this and suggests to Andy: ‘I am now going to label a third dice. You can then choose an arbitrary dice and I will take one of the remaining two.’ Can Beth label the third dice in such a way that she can always choose a dice with a better chance of winning, i.e., so thatD1  D2  D3  D1, meaning that the relation  is not transitive?