There is statistical evidence for a dearth of short-period (Porb < 4h) black hole (BH) low mass X-ray binaries (LMXBs) in the Galaxy. At short periods accretion onto the central object (be it a BH) may become inefficient because the cooling timescale of the gas is greater than the accretion timescale (this is the well known ADAF model). The nature of the switch is important in terms of the outburst timescales of transient sources. The switch may be sharp or occur smoothly over time. I show that the dearth can be explained if the switch to inefficiency occurs sharply at some fraction of the Eddington luminosity of the BH (fLEdd).