  In this paper we analyze the probability distributions associated with
rolling (possibly unfair) dice infinitely often. Specifically, given a
$q$-sided die, if $x_i\in\{0,\ldots,q-1\}$ denotes the outcome of the
$i^{\text{th}}$ toss, then the distribution function is $F(x)=\mathbb{P}[X\leq
x]$, where $X = \sum_{i=1}^\infty x_i q^{-i}$. We show that $F$ is singular and
establish a piecewise linear, iterative construction for it. We investigate two
ways of comparing $F$ to the fair distribution -- one using supremum norms and
another using arclength. In the case of coin flips, we also address the case
where each independent flip could come from a different distribution. In part,
this work aims to address outstanding claims in the literature on Bernoulli
schemes. The results herein are motivated by emerging needs, desires, and
opportunities in computation to leverage physical stochasticity in
microelectronic devices for random number generation.
