Denote by A(x)={a:ℝa’τxℝ•h} a circle zone on the three-dimensional sphere surface for each given h>0. For a given integer m, the authors investigate how many zones chosen randomly are needed to contain at least one of the points on the sphere surface m times. As an application, the lifetime of a sphere roller is investigated. The authors present empirical formulas for the mean, standard deviation and distribution of the lifetime of the sphere roller. Furthermore, some limit behaviors of the above stopping time are obtained, such as the limit distribution, the law of the iterated logiarithm, and the upper and lower bounds of the tail probability with the same convergent order.
