  Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of
characteristic coprime to 2, 3, and $p$, which also contains the primitive
$p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer and
Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the
sizes of prime Selmer groups over a family of cyclic prime twists of
non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of
mild constraints conforms to the distribution conjectured by Bhargava, Kane,
Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in
proving these results are the Riemann hypothesis over global function fields,
the Erd\"os-Kac theorem, and the geometric ergodicity of Markov chains.
