Summary: Let \(X_ 1,X_ 2,\dots,X_ n\) be a sequence of \(n\) random variables taking values in the \(\xi\)-letter alphabet set \(\Lambda=\{a_ 1,a_ 2,\dots,a_ \xi\}\). We consider the number \(N=N(n,k)\) of non-overlapping occurrences of a fixed \(k\)-letter word under (a) i.i.d. and (b) stationary Markovian hypotheses on the sequence \(\{X_ j\}_{1\leq j\leq n}\), and use the Stein-Chen method to obtain Poisson approximations for the same. In each case, results and couplings from A. D. Barbour, L. Holst and S. Janson [Poisson approximation (1992; Zbl 0746.60002)] are used to show that the total variation distance between the distribution of \(N\) and that of an appropriate Poisson random variable is of order (roughly) \(O(kS_{(k)})\), where \(S_{(k)}\) denotes the stationary probability of the word in question. These results vastly improve on the approximations obtained by A. P. Godbole [Adv. Appl. Probab. 23, No. 4, 851-865 (1991; Zbl 0751.60018)]. In the Markov case, we also make use of recently obtained eigenvalue bounds on convergence to stationarity due to P. Diaconis and D. Stroock [Ann. Appl. Probab. 1, No. 1, 36-61 (1991; Zbl 0731.60061)] and J. A. Fill [ibid. 1, No. 1, 62-87 (1991; Zbl 0726.60069)].