Let $K = \mathbb{F}_q(t)$ be the global function field of characteristic coprime to $2$ and $3$. Let $E$ be a non-isotrivial elliptic curve over $K$. Fix a prime number $l$ such that the primitive $l$-th roots of unity $\mu_l$ is contained in $\mathbb{F}_q$. Let $L$ be a $\mathbb{Z}/l\mathbb{Z}$ geometric Galois extension over $K$. We will focus on utilizing the Grothendieck-Lefschetz trace formula to compute a lower bound on the probability that the rank of $E(L)$ is equal to the rank of $E(K)$. If time allows, we will also explore how one can combine a probabilistic approach to obtain new geometric insights on the cohomology groups of some Hurwitz spaces.