# Rule of Tangent for Win-By-Two Games

Reza D. Noubary 1  and Drue Coles 2
• 1 Bloomsburg University
• 2 Bloomsburg University

Our study of win-by-two tie games is motivated by a famous 2010 Wimbledon tennis match whose final set was decided by the improbable score of 70-68. We introduce a trigonometric interpretation of the odds of winning points and games in tennis when serving from deuce. We place this result in the more general setting of a gambler’s ruin problem and also propose a performance measure to quantify the serving and receiving skill of one player relative to another. Then we extend the analysis to table tennis and volleyball. These latter games are similar to tennis in that the winner must obtain a certain minimum score while leading by two points, but they differ in their determination of which player serves a given rally and in whether a point is awarded to the receiver for winning a rally. We quantify the impact of these differences on the outcomes of games, assuming that the probability for a player to win a single point does not change during a game. We also apply a Markov chain analysis to arrive at our earlier results for tennis and to calculate the expected length of a game after reaching deuce. Finally, we develop the idea of “equivalent games” so that the analysis can be carried out using only the probability of winning a point (that is, without regard for the question of which player is serving).

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