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This paper presents improved approximation algorithms for the problem of multiprocessor scheduling under uncertainty (SUU), in which the execution of each job may fail probabilistically. This problem is motivated by the increasing use of distributed computing to handle large, computationally intensive tasks. In the SUU problem we are given n unit-length jobs and m machines, a directed acyclic graph G of precedence constraints among jobs, and unrelated failure probabilities qij for each job j when executed on machine i for a single timestep. Our goal is to find a schedule that minimizes the expected makespan. Lin and Rajaraman gave the first approximations for this NP-hard problem for the special cases of independent jobs, precedence constraints forming disjoint chains, and precedence constraints forming trees. In this paper, we present asymptotically better approximation algorithms. In particular, we improve upon the previously best O(log n)-approximation, giving an O(log log(min {m,n}))-approximation in the case of independent jobs. We also give an O(log(n+m) log log(min{m,n}))-approximation algorithm for precedence constraints that form disjoint chains (improving on the previously best O(log(n)log(m) log(n+m) over log log(n+m)-approximation by a (log n/log log n)2 factor when n = m Θ(1)). Our algorithm for precedence constraints forming chains can also be used as a component for precedence constraints forming trees, yielding a similar improvement over the previously best algorithms for trees. Our techniques include reducing SUU to a problem in stochastic scheduling, where machines must process a set of jobs with randomly distributed lengths. We show that our algorithms for (SUU) apply to a standard problem in this setting, giving the first approximation algorithms for preemptive stochastic scheduling on unrelated machines.