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We investigate the behavior of the sum and difference sets of $A \subseteq\mathbb{Z}/n\mathbb{Z}$ chosen independently and randomly according to abinomial parameter $p(n) = o(1)$. We show that for rapidly decaying $p(n)$, $A$is almost surely difference-dominated as $n \to \infty$, but for slowlydecaying $p(n)$, $A$ is almost surely balanced as $n \to \infty$, with acontinuous phase transition as $p(n)$ crosses a critical threshold.Specifically, we show that if $p(n) = o(n^{-1/2})$, then $|A-A|/|A+A|$converges to $2$ almost surely as $n \to \infty$ and if $p(n) = c \cdotn^{-1/2}$, then $|A-A|/|A+A|$ converges to $1+\exp(-c^2/2)$ almost surely as $n\to \infty$. In these cases, we modify the arguments of Hegarty and Miller onsubsets of $\mathbb{Z}$ to prove our results. When $\sqrt{\log n} \cdotn^{-1/2} = o(p(n))$, we prove that $|A-A| = |A+A| = n$ almost surely as $n \to\infty$ if some additional restrictions are placed on $n$. In this case, thebehavior is drastically different from that of subsets of $\mathbb{Z}$ and newtechnical issues arise, so a novel approach is needed. When $n^{-1/2} =o(p(n))$ and $p(n) = o(\sqrt{ \log n} \cdot n^{-1/2})$, the behavior of $|A+A|$and $|A-A|$ is markedly different and suggests an avenue for further study.These results establish a "correspondence principle" with the existing resultsof Hegarty, Miller, and Vissuet. As $p(n)$ decays more rapidly, the behavior ofsubsets of $\mathbb{Z}/n\mathbb{Z}$ approaches the behavior of subsets of$\mathbb{Z}$ shown by Hegarty and Miller. Moreover, as $p(n)$ decays moreslowly, the behavior of subsets of $\mathbb{Z}/n\mathbb{Z}$ approaches thebehavior shown by Miller and Vissuet in the case where $p(n) = 1/2$.

When almost all sets are difference dominated in $\mathbb{Z}/n\mathbb{Z}$