Estimating probabilities on a pedigree by dependent samples, namely realizations of a Markov chain, has been explored as an alternative method when exact computation is not feasible. If the transition kernel of the Markov chain is aperiodic and irreducible, convergence of the estimates to the true probabilities is guaranteed by the ergodic theorem. However, reducibility is a potential problem for genetic pedigree analysis unless the Markov chain is constructed appropriately. In the present paper, we propose a scheme for constructing an irreducible Markov chain for pedigree data. Transitions between communicating classes, which can be found explicitly, are made by using a Metropolis jumping kernel. The method has been demonstrated to be much more efficient than other currently existing methods.