The purpose of this short note is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain $Ω$ by the largest mean first exit time of the associated drift-diffusion process via $λ_1 ≥ \frac{1}{\sup_{x \in \Omega} 𝔼_x τ_{Ω^c}}.$ Instead of looking at the mean of the first exist time, we study quantiles: let $d_{p, 𝟃 Ω}:Ω → ℝ_{≥ 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $λ_1 ≥ \frac{\log{(1/p)}}{\sup_{x 𝟄 Ω} d_{p,𝟃 Ω}(x)}.$ Moreover, as $p → 0$, this lower bound converges to $λ_1$.