Discrete distributions have played an important role in the reliability theory. In order to obtain Bayes estimators, researchers have adopted various conventional techniques. Generalizing the results of Maiti (1995), Chaturvadi and Tomer (2002) dealt with the problem of estimating P{X1, X2, …, XkY}, where random variables X and Y were assumed to follow a negative binomial distribution. Agit et al. obtained Bayesian estimates of the reliability functions and P{X1, X2, …, XkY} considering X and Y following binomial and Poisson distributions. The reliability function of the generalized Poisson and generalized geometric distribution is investigated. The expression for P{X1, X2, …, XkY} was obtained with X’s and Y following a Poisson distribution and some particular cases are shown.