Stochastic dynamical systems have emerged as fundamental models across numerous application domains, providing powerful mathematical representations for capturing uncertain system behavior. In this paper, we address the problem of runtime safety and reach-avoid probability prediction for discrete-time stochastic systems with online observations, i.e., estimating the probability that the system satisfies a given safety or reach-avoid specification. Unlike traditional approaches that rely solely on offline models, we propose a framework that incorporates real-time observations to dynamically refine probability estimates for safety and reach-avoid events. By introducing observation-aware barrier functions, our method adaptively updates probability bounds as new observations are collected, combining efficient offline computation with online backward iteration. This approach enables rigorous and responsive prediction of safety and reach-avoid probabilities under uncertainty. In addition to the theoretical guarantees, experimental results on benchmark systems demonstrate the practical effectiveness of the proposed method.