Properties of dynamical systems with dichotomous Markov noise which exhibit crises are investigated. We find numerically the dependence of the mean residence time on the precrisis attractor on the transition rate (or transition probability in the discrete-time case) of dichotomous Markov noise. To explain this dependence, we construct a simple Markov chain model, which allows us to find the mean residence time for the given transition rate with a good accuracy. Next, we find the distribution of residence times for a system driven by dichotomous Markov noise and also build a simple model to explain its properties.