  We revisit the probabilistic construction of sparse random matrices where
each column has a fixed number of nonzeros whose row indices are drawn
uniformly at random. These matrices have a one-to-one correspondence with the
adjacency matrices of lossless expander graphs. We present tail bounds on the
probability that the cardinality of the set of neighbors for these graphs will
be less than the expected value. The bounds are derived through the analysis of
collisions in unions of sets using a {\em dyadic splitting} technique. This
analysis led to the derivation of better constants that allow for quantitative
theorems on existence of lossless expander graphs and hence the sparse random
matrices we consider and also quantitative compressed sensing sampling theorems
when using sparse non mean-zero measurement matrices.
