Green’s Dictionary of Slang

fair shake of the dice phr.

(Aus., N.Z.) a phr. meaning ‘be fair, give us a chance’.

[NZ]McGill Dict. of Kiwi Sl. 44/2: fair shake (of the dice) [...] fair whack are some more appeals for fairness.
[NZ]McGill Reed Dict. of N.Z. Sl. [as cit. 1988].
[Aus]Macquarie Dict. [Internet] fair shake of the dice, an appeal for fairness or reason.