The statistical properties of charge exchanged between an open system and its environment can be analyzed by studying jumps in the system state along dynamical trajectories. Two complementary approaches are Full Counting Statistics, where the distribution of exchanged charge is studied at a fixed time, and First Passage Statistics, which studies the distribution of the time taken to reach a fixed amount of charge. These two approaches are closely related. For both current-like observables and dynamical observables, we discuss the relationship that usually connects both approaches, and then identify situations in which the relationship can be violated. We find that these violations indicate the presence of meta-stable or dark states. We also discuss generalizations of the theory to finite times, to more general counting observables, and beyond Markovian dynamics.