  The goal of this paper is to investigate the missing part of the story about
the relationship between the orthogonal polynomial ensembles and Painlev\'e
equations. Namely, we consider the $q$-Racah polynomial ensemble and show that
the one-interval gap probabilities in this case can be expressed through a
solution of the discrete $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ equation.
Our approach also gives a new Lax pair for this equation. This Lax pair has an
interesting additional involutive symmetry structure.
