(a) Average number of lensed images magnified by more than ﹩A﹩ that will be observed in the inner ﹩2\theta _{\mathrm{E}\,}﹩ with a limiting ﹩K﹩‐band magnitude ﹩K_{0}﹩, for ﹩D_{s}=2R_{0}﹩ and the stellar lens surface density model of eq. (11) with ﹩\alpha =\frac{3}{2}﹩. (b) The fraction of time that at least one lensed image magnified by more than ﹩A﹩, will be observed in the inner ﹩2\theta _{\mathrm{E}\,}﹩ with a limiting ﹩K﹩‐band magnitude ﹩K_{0}﹩ (﹩p\ll 1﹩ scales linearly with ﹩\hat{\Sigma }_{s}﹩). The labeled solid lines are for the BH and stars and the unlabeled dotted lines, which follow the same ﹩K_{0}﹩ order, are for the BH alone.