This article has two purposes: (a) To construct useful statistics by estimating important probabilities, a method that could easily be used in a first course in mathematical statistics as an alternative to standard methods (like the likelihood ratio). (b) To discuss these probability estimates as another method for reporting the findings of a quantitative stujdy. Since this idea has been used in certain selected nonparametric problems, it is not new. Nevertheless, its generality has not been explored, and its usefulness in reporting data has been seemingly overlooked. The basic idea is to consider one or more important probabilities that are associated with the problem under consideratiori and then to estimate these probabilities, assuming the most reasonable mathematical model available. We believe that these estimates, along with appropriate error bounds, will be as useful to the consumers of statistical analyses as are our present techniques. As a matter of fact, it would be easy, in a first course in mathematical statistics, to use this approach, as it produces nearly all of the standard parametric and nonparametric techniques associated with one or two distributions. Since the main purpose of our paper is to advertise this approach, we give several illustrations which demonstrate its adaptability, without going into any one in great depth. It is hoped that many statisticians, including the authors, will want to explore its various characteristics further.