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Current stochastic model checkers do not make counterexamples for property violations readily available. In this paper, we apply directed explicit state-space search to discrete and continuous-time Markov chains in order to compute counterexamples for the violation of PCTL or CSL properties. Directed explicit state-space search algorithms explore the state space on-the-fly, which makes our method very efficient and highly scalable. They can also be guided using heuristics which usually improve the performance of the method. Counterexamples provided by our method have two important properties. First, they include those traces which contribute the greatest amount of probability to the property violation. Hence, they show the most probable offending execution scenarios of the system. Second, the obtained counterexamples tend to be small. Hence, they can be effectively analyzed by a human user. Both properties make the counterexamples obtained by our method very useful for debugging purposes. We implemented our method based on the stochastic model checker PRISM and applied it to a number of case studies in order to illustrate its applicability. ; published