  In this paper, we consider the deformed Fredholm determinant of the confluent
hypergeometric kernel. This determinant represents the gap probability of the
corresponding determinantal point process where each particle is removed
independently with probability $1- \gamma$, $0 \leq \gamma <1$. We derive
asymptotics of the deformed Fredholm determinant when the gap interval tends to
infinity, up to and including the constant term. As an application of our
results, we establish a central limit theorem for the eigenvalue counting
function and a global rigidity upper bound for its maximum deviation.
