Volleyball scoring systems

1. INTRODUCTION The widespread interest in sports in our culture provides a great opportunity to catch students' attention in mathematics and statistics classes. Many students, whose eyes would normally glaze over after ten minutes of algebra, will happily spend hours analyzing their favorite sport. In recent years the mathematical literature on sports (e.g. [1] and [2]) has expanded greatly and covers a broad spectrum of sports, including baseball, football, soccer, tennis, and track and field. Volleyball, an increasingly popular sport, has a mathematically interesting scoring system. The purpose of this paper is to show students, through an analysis of volleyball scoring systems, how we can model volleyball games as Markov chains, obtaining formulas for the proportion of points won by each team in the long run. We can also calculate the exact probabilities of winning a point, a game and a match for each scoring system. Under careful guidance of a faculty instructor, this paper can be well appreciated by college or high school students who have finished a first probability and statistics course. This study may also be of interest to volleyball players, coaches and fans, since the USA Volleyball official rules for scoring were changed in 1999. Our model predicts that the outcome of volleyball matches will rarely be affected by the change in scoring systems. 2. VOLLEYBALL SCORING There are two different scoring systems commonly used in volleyball: side-out scoring and rally point scoring. In both systems, a rally begins with a serve by one of the two teams and ends when the ball touches the floor or goes out, or a team commits a fault or fails to return the ball to the opponent's court. The team that wins the rally gains the right to serve in the next rally. Traditionally, side-out scoring has been used. This means that a point is awarded only when the serving team wins the rally. When the receiving team wins the rally it gains the right to serve next, but is not awarded a point. Recently, USA Volleyball (USAV) has followed the International Volleyball Federation (FIVB) in moving to rally point scoring. In rally point scoring a point is awarded after each rally to the team that won the rally. Games are played either to 15 or 25 points. If there is no scoring cap, then the winning team must be ahead by at least two points. For example, in a 15 point game, if the score reaches 15-14 or 14-15, the game goes on indefinitely until one of the two teams is two points ahead. If there is a scoring cap, a team's score cannot exceed 17. …