The Markov decision process model is a powerful tool in planing tasks and sequential decision making problems. The randomness of state transitions and rewards implies that the performance of a policy is often stochastic. In contrast to the standard approach that studies the expected performance, we consider the policy that maximizes the probability of achieving a pre-determined target performance, a criterion we term *probabilistic goal Markov decision processes*. We show that this problem is NP-hard, but can be solved using a pseudo-polynomial algorithm. We further consider a variant dubbed ``chance-constraint Markov decision problems," that treats the probability of achieving target performance as a constraint instead of the maximizing objective. We show that this variant is NP-hard, but can be solved in pseudo-polynomial time.