  As a precursor to studying the bound states of multiple non-extremal black
holes in $\mathcal{N}=2$, $d=4$ supergravity, we investigate the stability of a
near-extremal D0-D4 black hole in the probe limit, when the parameters of the
black hole solution lie within a certain regime. We determine whether it is
possible to form bound states of this "core" non-extremal black hole with BPS
probe particles, and whether it is possible for the "core" black hole to decay
by the emission of such BPS probes either to a local minimum of the probe
potential, or spatial infinity. We first carry out a qualitative analysis of
the probe potential to determine when quantum tunneling of probes from the
black hole is possible. We then find the wavefunction of the scattered probe by
using the WKB approximation to solve the Dirac equation in the black hole
background, and use this solution to compute the tunneling amplitude.
