Mathematical Exploration
“Calculating the impact of change in probability of winning a point on the probability of winning the match.”
Candidate Name: Rohan Ketan Dhamsania
Candidate number: 003436-0017
Session: November 2014
School: The Galaxy School
Mathematics SL
INTRODUCTION
Tennis is a game, which rewards skill and technique. The roll of ‘winning by chance’ is minimized in tennis, in fact Pete Sampras, a tennis legend, had said, “The difference of great players is at a certain point in a match they raise their level of play and maintain it.” - it is the probability of the point that matters the most. I have been playing tennis since a tender age of 10 years and have experienced it first hand that
However, initially I was not sure as to how would the probability of winning a point be applied to that of match. After exploring and trying various methods like tree diagrams and binomial distribution, I found out that Binomial distribution was the most accurate and effective
Let us assume the probability to win a point to be 0.60 and calculate the probability of winning the match:
Table – 1 showing the calculations for player A winning a game: p = Probability of player A to win a point q = Probability of player A to lose a point (or player B to win a point) n = The number of points in which the player A can lose the required number of points (remember the last point has to be won by player A to win the game) r = The number of points lost by player A
HOW IS THE GAME WON p q n r nCr “a”[ ] WINNING CALCULATIONS (the values are substituted into the formula derived for P(winning a game to x points) “a” WIN
Game to 0 0.60 0.40 4 0 4C0 = 1 1(0.60)^4 (0.40)^0 0.129600
Game to 15 0.60 0.40 4 1 4C1= 4 4(0.60)^4 (0.40)^1 0.207360
Game to 30 0.60 0.40 5 2 5C2= 10 10(0.60)^4 (0.40)^2 0.207360
Game after deuce 0.60 0.40 6 3 6C3= 20 20〖(0.60)〗^5 〖(0.40)〗^3 {1/(1-2(0.60)(0.40))} 0.191409
Total probability of winning the game is the summation of above all solutions 0.735609
Table – 2 showing the calculations for the server winning a set:
a = Probability of winning n number of games for player A b = Probability of losing n number of games for player A (or winning n number of games for player
Tennis serve Introduction to tennis Tennis is a sport that can be played individually against a single opponent (singles) or between two teams of two players each (doubles). Each player uses a tennis racket that is strung with cord to strike a hollow rubber ball covered with felt over or around a net and into the opponent's court. The object of the game is to play the ball in such a way that the opponent is not able to play a valid return. The player who is unable to return the ball will not gain a point, while the opposite player will.
The objective is to win each rally by serving or returning the ball so the opponent is unable to keep the ball in play. A rally is over when a player (or team in doubles) is unable to hit the ball before it touches the floor twice, is unable to return the ball in such a manner that it touches the front wall before it touches the floor, or when a hinder is called.
Our conclusion is that while a rise in each stat had some affect in the rise or fall of winning percentage, we could not determine a single stat that had a direct affect on the dependent variable (Winning Percentage). Our results were more effective when we ran the test on how the combination of all stats affected winning percentage, however, this would be obvious given the nature of our study.
the last game that player will ever participate in. On the other hand the player could also go back in and
The last part of coaching tennis is learning how to keep score. The score of the person that is serving is always the first number to be called out such as 15-love. Scoring goes: love, 15, 30, 40, game. If you and your opponent get to 40-40 that is called deuce. From there you go deuce, advantage server or advantage receiver, and then if the person who had the advantage wins the next point then the game is over, but if the receiver wins the point than you go back to deuce. One set is six games, and a full match is the best of six sets.
you throw your opponent the more points you receive. You can win if you throw
Years of playing the game and not improving, Gawande incidentally finds himself play tennis with a young man who is a tennis couch. The young man gives Gawande a tip about keeping his feet under his body when hitting the ball. At first he is uncertain, stating, “My serve had always been the best part of my game….. With a few minutes of tinkering, he’d added at least ten miles an hour to my serve. I was serving harder than I ever had in my life” (Gawande, 2011, p.3).
▶When you are in the low-probability zone with a low win ratio, you are statistically more likely to experience more consecutive losses.
Table 1.1 shows the probabilities of a starting hand. Which means that it shows the probability, in percentage, of getting a certain card handed out in the first round. So the probability of which card one player gets.
...one team is completely annihilated or they surrender. After game is called, a cheer rises from the winning team. The teams that were just playing try to figure out who shot who and where everyone was located at. You also compare "wounds".
These ideas are useful in many contexts, including sports, and they allow us to make sense of events and ourselves. Reference: 3.2 Independent and Mutually Exclusive Events Statistics | OpenStax. n.d. - n.d. - n.d. Openstax.org - a free website.
When luck is with you, you can win in spite of low chance of winning; when luck is not with you, you could 12 fail even with a good chance of winning. The hot-hand fallacy and gamblers’ fallacy are assumed to be common among gamblers because it is thought that they have a strong tendency to believe that outcomes for future bets are predictable from those of previous ones. In chapter 4, a mechanism of the gamblers' fallacy creating the hothand effect will be revealed. Belief in a hot-hand is “If you have been winning, you are more likely to win again.” The term “hot hand” was initially used in basketball to describe a basketball player who had been very successful in scoring over a short period. It was believed that such a player had a “hot hand” and that other players should pass the ball to him to score more. This term is now used more generally to describe someone who is winning persistently and can be regarded as “in luck”. In gambling scenarios, a player with a genuine hot hand should keep betting and bet more. There have been extensive discussions about the existence of the hot hand effect. Some researchers have failed to find any evidence of such an effect (Gilovich, Vallone and Tversky, 1985)Others claim there is evidence of the hot hand effect in games that require considerable physical skill, such as golf, darts, and basketball (Gilden and Wilson, 1995; Arkes, 2011; Yaari and Eisenmann, 2011). People gambling on sports outcomes may continue to do so after winning because they believe they have a hot hand. Such a belief may be a fallacy. It is, however, possible that their belief is reasonable. For example, on some occasions, they may realize that their betting strategy is producing profits and that it would be sensible to continue with it. Alternatively, a hot hand could arise from some change in their betting strategy. For example, after winning, they may modify their bets in 13 some way to increase their
Two teams of eleven players each participate in getting the ball into the other team’s goal, thus scoring a goal. The team that has scored more goals at the end of the game wins. If both teams have scored an equal number of goals, then the game is a tie. Each team is controlled by a captain. In game play, players make an effort to create goal scoring occasions through individual control of the ball, such as dribbling, passing the ball to a team-mate, and by taking shots at the goal, that is guarded by the goalkeeper belonging to the other team....
The margins for success and failure as a world-class athlete can be miniscule. Skiers go wide on the third gate of a downhill race to find they have not only lost the gold medal, but any medal. Members of the PGA, after playing 72 holes, find themselves losing the tournament by one stroke, as a result of the missed three-foot putt on the second day of competition.