Our purpose in this paper is to obtain the \emph{exact distribution} of the number of buffer starvations within a sequence of $N$ consecutive packet arrivals. The buffer is modeled as an M/M/1 queue. When the buffer is empty, the service restarts after a certain amount of packets are \emph{prefetched}. With this goal, we propose two approaches, one of which is based on \emph{Ballot theorem}, and the other uses recursive equations. The Ballot theorem approach gives an explicit solution, but at the cost of the high complexity order in certain circumstances. The recursive approach, though not offering an explicit result, needs fewer computations. We further propose a fluid analysis of starvation probability on the file level, given the distribution of file size and the traffic intensity. The starvation probabilities of this paper have many potential applications. We apply them to optimize the quality of experience (QoE) of media streaming service, by exploiting the tradeoff between the start-up delay and the starvation.