Thus far, judging the fate of a massive star (either a neutron star [NS]or a black hole) solely by its structure prior to core collapse has been ambiguous. Our work and previous attempts find a nonmonotonic variation of successful and failed supernovae with zero-age main-sequence mass, for which no single structural parameter can serve as a good predictive measure. However, we identify two parameters computed from the pre-collapse structure of the progenitor, which in combination allow for a clear separation of exploding and nonexploding cases with only afew exceptions (∼1%-2.5%) in our set of 621 investigated stellar models. One parameter is M 4 , defining the normalized enclosed mass for a dimensionless entropy per nucleon of s=4, and the other is dm M dr 1000 km s, being the normalized massderivative at this location. The two parameters μ 4 and M 4 μ 4 can be directly linked to the mass-infall rate, Ṁ , of the collapsing star and the electron-type neutrino luminosity of the accreting proto-NS, L M M ns eμ n , which play a crucial role in the "critical luminosity" concept for the theoretical description of neutrino-driven explosions as runaway phenomenaof the stalled accretion shock. All models were evolved employing the approach of Uglianoetal. for simulating neutrino-driven explosions in spherical symmetry. The neutrino emission of the accretion layer is approximated by a gray transport solver, while the uncertain neutrino emission of the 1.1 M e proto-NS core is parameterized by an analytic model. The free parameters connected to the core-boundary prescription are calibrated to reproduce the observables of SN1987A for five different progenitor models.