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Fault tree (FT) is a standardized notation for representing relationships between a system's reliability and the faults and/or the events associated with it. However, the existing FT fault models are only capable of portraying permanent events in the system. This is a major hindrance since these models fail to reflect accurately the other classes of faults, such as soft-faults, which are often temporary events that usually disappear after the source of the interference is no longer present. This paper proposes a new fault tree modeling paradigm, to capture the impact of temporal events in systems, called temporal dynamic fault trees (TDFTs). TDFTs are utilized to model the characteristics and dependencies between different temporal events, soft-faults, and permanent faults. These features are integrated into the proposed probabilistic models of the temporal gates, which are modeled as priced-timed automata. This paper also proposes a new FT analysis methodology, based on statistical model checking, designed to circumvent the state-explosion problem that is inherent to other model-checking approaches. The proposed analysis is able to evaluate the impact of temporal faults in systems, as well as to estimate the reliability and availability of the system over extended periods of time. The experiments reported in this paper demonstrate the versatility and scalability of the proposed approach. For instance, the results display the impact that temporal events may have in a digital system. Our observations indicate that while regular soft-fault analyses tend to underestimate metrics such as system reliability, TDFT analysis shows remarkable consistency with radiation testing, with differences of under 2%, in the conducted analysis.