Let Xı, X .. be a sequence of Bernoulli trials governed by a homogeneous third-order lwo-state Markov chain. The probabilirty function of the number of occurrences in n succes- sive trials, is obtained. In addition to this, assuming that steady State is already attained the li- miting function of is obtained under the condition that nP (Xj = 1) = u as n 00, Finally we can note that this limiting probabüity function can be generalized in terms of Leguerre poly- nomials as already shown in the relevant literatüre.