Run statistics about a sequence of independent geometrically distributed random variables has attracted some attention recently in many areas such as applied probability, reliability, statistical process control, and computer science. In this paper, we first study the mean and variance of the number of alternating runs in a sequence of independent geometrically distributed random variables. Then, using the relation between the model of geometrically distributed random variables and the model of random permutation, we can obtain the variance in a random permutation, which is difficult to derive directly. Moreover, using the central limit theorem for dependent random variables, we can obtain the distribution of the number of alternating runs in a random permutation.