From the literature, it is known that solutions of homogenous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. In this paper we take a probabilistic standpoint to analyze these phenomena by assuming that both the initial conditions and the coefficients of the equation have random nature. Under these assumptions we find the probability for deviations to occur, which turns out very close to unity even for equations of low degree, which means that peak is typical. We also address other issues such as evaluation of the mean magnitude and maxumum value of peak.