Many statistical problems can be formulated as discrete missing data problems (MDPs). Examples include change-point problems, capture and recapture models, sample survey with non-response, zero-inflated Poisson models, medical screening/diagnostic tests and bioassay. This paper proposes an exact non-iterative sampling algorithm to obtain independently and identically distributed (i.i.d.) samples from posterior distribution in discrete MDPs. The new algorithm is essentially a conditional sampling, thus completely avoiding problems of convergence and slow convergence in iterative algorithms such as Markov chain Monte Carlo. Different from the general inverse Bayes formulae (IBF) sampler of Tan, Tian and Ng (Statistica Sinica, 13, 2003, 625), the implementation of the new algorithm requires neither the expectation maximization nor the sampling importance resampling algorithms. The key idea is to first utilize the sampling-wise IBF to derive the conditional distribution of the missing data given the observed data, and then to draw i.i.d. samples from the complete-data posterior distribution. We first illustrate the method with a performing example and then apply the method to contingency tables with one supplemental margin for an human immunodeficiency virus study.