Tennis is a universal sport. Like any other sport, tennis has its own terminology and scoring system. During a match broadcast on television, viewers usually have the opportunity to see various statistics. These statistics would be the number of aces, first and second serve percentages, number of winners and errors, etc. The average viewer would only go so far as to know who won, but these statistics provide a much deeper understanding of many underlying aspects of the match, such as how a player won and the odds of winning. Meanwhile, the construction of probability formulas is present due to the hierarchical structure of the scoring system. For example, points are earned throughout service games; these points are interwoven into the sets, as well as the match. So, tennis is a mathematical game that depends heavily on the players' chances of winning the match2. Scoring a Tennis MatchTennis has a distinctive scoring system that can be confusing for many beginners or people who are simply unfamiliar with the sport. However, the basic scoring system involves having points required to win games, games required to win sets, and sets required to win the match.2.1 GamesThe score of a game usually starts from 0 ( love), then as points are earned, they gradually increase and become 15, 30, 40 and then game respectively. If both players reach 40, this is called "two". With two people, whoever wins the next point gets an “advantage”. If the advantaged player wins the next point, it will be a game; however, if they lose the next point, they come back to two. Throughout the game, the server remains constant and will alternate between games. (Figure 2.1.1) This figure is him...... middle of paper ....... Therefore, the probability that the server wins the game without reaching two occurs with probability p^4+ 4p ^4 q + 10p^4 q^2= p^4 (1 + 4q + 10q^2). The probability of the score reaching two occurs with probability 20 p^3 q^3 (since there are 20 paths that can occur to reach two points and each player gains exactly 3 points). The probability that the server wins the game from two: To win the game from two, the server must win the next two points. This happens with probabilityp^2. If the server wins the next point out of two followed by the receiver winning one point, or if the receiver wins the next point out of two followed by the server winning one point, then the score becomes two. This happens with a probability of 2pq. If d is the probability that the server wins the game when the score is two, then d = p^2+ 2pqd. Solving for d gives d=p^2/(1-2pq)