The diagnosis problem amounts to deciding whether some specific "fault" event occurred or not in a system, given the observations collected on a run of this system. This system is then diagnosable if the fault can always be detected, and the active diagnosis problem consists in controlling the system in order to ensure its diagnosability. We consider here a stochastic framework for this problem: once a control is selected, the system becomes a stochastic process. In this setting, the active

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... nosis problem consists in deciding whether there exists some observationbased strategy that makes the system diagnosable with probability one. We prove that this problem is EXPTIME-complete, and that the active diagnosis strategies are belief-based. The safe active diagnosis problem is similar, but aims at enforcing diagnosability while preserving a positive probability to non faulty runs, i.e. without enforcing the occurrence of a fault. We prove that this problem requires non belief-based strategies, and that it is undecidable. However, it belongs to NEXPTIME when restricted to belief-based strategies. Our work also refines the decidability/undecidability frontier for verification problems on partially observed Markov decision processes.