## Honors Theses, 1963-2015

Thesis

1991

Mathematics

Philip Byrne

#### Abstract

This paper studies the probabilities within the game Jai-alai. Before explaining the probabilities, one first needs to understand the game itself. Jai-alai is a racquetball-like game where eight ordered players compete against one another in the following manner: Player 1 plays Player 2. The winner of this game plays Player 3, while the loser goes to the end of the line behind Player 8. In the first seven games, 1 versus 2," versus 3,"""? versus 8, the winner is awarded one point, and in every game after that two points. A player wins the match when he is the first to acquire seven or more points. The question involving probabilities is the following: Suppose all players are of equal ability. Is there any advantage to being in the front of the line (like players 1 and 2) versus being in the end of the line? Another way of restating the problem is calculate the probabilities each player has of winning the match given that they all are of equal ability.
Two approaches are used towards solving this problem. The first method involves running simulations on a computer and tabulating these results. Then using statistics a range can be set in which the probabilities are bounded. The second approach involves modeling the game by a Markov Chain. The Markov Chain can be written in matrix form, which in turn can be manipulated so that the probabilities of the players winning the game can be calculated exactly. Also in this second approach, it was necessary to look at smaller player number versions of the Jai-alai game to detect patterns that would be in the regular Eight Player Game. Each of these methods has advantages and drawbacks. The advantage of the simulation is that it can give a fair estimate of the probabilities in a short amount of time, while the advantage of the Markov Chain is that it can give the exact probabilities.

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