Ross A basic probability identity is established and applied to obtain new simulation estimators concerning (a) system reliability, (b) a multi-valued system, and (c) the probability distribution of the time until a given pattern occurs when data is generated by a stationary Markov chain. It has been shown that the variance of this new estimator is often of the order a-squared when the usual estimator's variance is of order a and a is small. The investigator will indicate how this estimator can be combined with standard variance reduction techniques. Additional studies of the estimator, as well as on variations of it are proposed as are additional applications. The research also concerns how the identity can be used to provide approximations and bounds in applied probability. General bounds as well as specific applications are indicated and proposed for further study. Many problems involving chance phenomena are too complex for one to obtain analytical solutions. In such cases a combination of a probabilistic analysis and a computer simulation leading to statistical estimators of the quantities of interest is often a potent method of attack. The investigator has indicated how a basic probability identity can be effectively employed in obtaining efficient simulation estimators. The purpose of this research is to study its usefulness, both in the simulation and the probabilistic analysis phase, in a variety of applications. The investigator has also indicated extensions of the identity along with potential areas of applications. ***
