  In this paper, we address the question: What is the probability of
stellar-mass black hole (BH) binaries co-existing in a globular cluster with an
intermediate-mass black hole (IMBH)? Our results suggest that the detection of
one or more BH binaries can strongly constrain the presence of an IMBH in most
Galactic globular clusters. More specifically, the detection of one or more BH
binaries could strongly indicate against the presence of an IMBH more massive
than $\gtrsim 10^3$ M$_{\rm \odot}$ in roughly 80\% of the clusters in our
sample. To illustrate this, we use a combination of N-body simulations and
analytic methods to weigh the rate of formation of BH binaries against their
ejection and/or disruption rate via strong gravitational interactions with the
central (most) massive BH.
  The eventual fate of a sub-population of stellar-mass BHs (with or without
binary companions) is for all BHs to be ejected from the cluster by the central
IMBH, leaving only the most massive stellar-mass BH behind to form a close
binary with the IMBH. During each phase of evolution, we discuss the rate of
inspiral of the central BH-BH pair as a function of both the properties of the
binary and its host cluster.
