A new probabilistic method is introduced which can be used to obtain a global map of the distribution of the velocity and force transmission characteristics of robot manipulators. The method yields the global distributions of volumes of the velocity and force ellipsoids. It utilizes the most basic definition of the properties of the mapping functions between the joint coordinates and the Cartesian (task) coordinates. The method is extremely simple, numerical in nature, and readily implemented on a computer. It requires very simple and limited analytical formulations. No inverse calculations are involved. The method can be utilized for optimal positioning of task spaces within workspaces of manipulators. Velocity and force transmission (alone and in combination) may be used as optimality criteria. Complex task space and manipulator workspace geometries are readily handled. The tasks, velocity, and force transmission requirements may assume any arbitrary, even conditional, distributions within the task space. These distributions, especially conditional distributions, are extremely difficult, if not impossible, to handle using other methods. The application of the method to task planning and optimal manipulator synthesis is discussed. Several examples are presented.