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
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).
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.
the last game that player will ever participate in. On the other hand the player could also go back in and
Imagine playing a six hour tennis match! Well, the longest match ever played was six hours long; played by Vicki Nelson and Jean Hepner in 1984. (Seminara, 2009) The scoring for tennis is kind of difficult if not familiar of how the game is played. The way the scoring goes is “luv, 15, 30, 40, or deuce if the score is 40-40. If it is 30-30, then it is said like “30 all”. In order to win, the game is played best of three matches. The winner must win by two games. In my study I will be getting the speeds of a group of tennis girls hitting the tennis balls. My hypothesis is that the more experience you have in tennis, the fast you should be able to hit. Because there are so many varieties of elements in tennis, one should consider the difference in racquets, tennis shoes, courts, and balls
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
▶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".
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.
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 probability of the score reaching deuce occurs with probability 20 p^3 q^3 (since there are 20 paths that can occur in reaching deuce and each player wins exactly 3 points).