On the winning probabilities and mean durations of volleyball

In this paper we consider the volleyball under the assumption that the probability of winning a single rally is independent of the other rallies and constant during the game. Fixing two parameters which indicate the probabilities of winning a rally for the serving team, we derive the exact expression of the probability of winning a set and a match, even in the case of sets ending with a tie break, in the present rally point- and in the former side-out scoring systems. Furthermore we are able to evaluate by a simple direct computation the mean duration of the sets in both the scoring systems, obtaining, as well known in the practice, that the change in the scoring system reduced the (expected) length of the matches. The analysis of the matches between the 4 top teams of the Italian Volley League in the period 2001-2012 shows that the model is adequate and accurate in describing the game.