The Bureau of Mines used mathematical reliability theory to define, for use in the mining industry, the concepts of risk, safety, reliability, hazard, and mean time between accidents. In this report, the definitions are explained theoretically and illustrated by application to actual underground mine accident data extending over a 9 1/2-year period. The theoretical development proceeds along two parallel analytical paths. For each line of reasoning, the data are presumed to be independent. If they are not, the analysis stops, although there are interesting implications when dependency is demonstrated. The first method uses the elapsed times between accidents as the basic data. These elapsed times are fitted to a negative exponential distribution function using least squares and yielding the function parameter, p1. The p1 is used to form the poisson distribution function for the data, which is then used to define the safety-related concepts. The second method uses the maximum likelihood estimator to formulate a poisson distribution function for the data. The function is tested for adequacy by a goodness-of-fit test; if the fit is acceptable, the poisson function is used to define the concepts. Both methods were applied to the data base, yielding nearly identical results.