We consider the event-triggered state estimation of a finite-state hidden Markov model with a general stochastic event-triggering condition. Utilizing the change of probability measure approach and the event-triggered measurement information available to the estimator, analytical expressions for the conditional probability distributions of the states are obtained, based on which the minimum mean square error event-based state estimates are further calculated. We show that the results also cover the case of packet dropout, under a special parameterization of the event-triggering conditions. With the results on state estimation, a closed-form expression of the average sensor-to-estimator communication rate is also presented. The effectiveness of the proposed results is illustrated by a numerical example and comparative simulations.