We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. When applied to graph Laplacians, this leads to ultrasparsifiers that in expectation behave as the nearly-optimal ones given by [Kolla-Makarychev-Saberi-Teng STOC‘10]. Combining this with the recursive preconditioning framework by [Spielman-Teng STOC‘04] and improved embedding algorithms, this leads to algorithms that solve symmetric diagonally dominant linear systems and electrical flow problems in expected time close to m log n . Part of this work was done while at CMU This work was partially supported by AFOSR Award FA9550-12-1-0175. Part of this work was done while at CMU and was supported by a Microsoft Research PhD Fellowship This work was partially supported by NSF grant CCF-1111257.