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Friday, March 18, 2016

Everyday Math: Sports

Game Time: Math Style




Known for underdog upsets and nail-biting game winners, the NCAA Basketball Tournament takes the cake for the most exciting annual college sports event. Nicknamed March Madness, the tournament begins with 64 teams, but after 6 rounds of games and plenty of Cinderella stories, the field is narrowed to only one National Champion. March Madness rallies the obvious sports fanatic, but also makes it easy for anyone to stay informed and enjoy the thrill that is collegiate sports. You may be thinking, "Okay.. but what does this have to do with math?". Staying with the trend of Everyday Math, the tournament inspired us to look into some key aspects of mathematics that are incorporated into various sporting events.


We'll start by taking a look at our inspiration- March Madness.



College Basketball

     Before the tournament tips off, it is estimated that more than 40 million Americans flock to ESPN or other sports-based websites to fill out their own tournament bracket. In all, it is estimated that about 70 million brackets will be completed. Bracket-filling strategy is a personal preference. Some stay loyal to their favorite college, selecting them to win-it-all, even though it isn't probable. Others hope to shock their competitors and pick underdogs, crossing their fingers that a rare bracket will do the trick. On the other hand, some people pick top seeded teams, leaning towards favorites. 



According to USA Today, the odds that you randomly select a perfect bracket– that's guessing every game correctly– are 1 to 9.2 quintillion. 
       More serious betters may look at the mathematics behind the tournament. Experts broke down mathematical models that can be used to simulate the tournament. The tournament can be thought of as a series of coin flips rather than games. However, instead of there being a 50/50 chance of flipping heads or tails, the odds are stacked differently depending on the teams playing. The NCAA has given each team a seed and has published data regarding the results of each of these seed numbers in the past. For an example, when #5 seeds and #12 seeds play against each other, the #5 seed wins 65% of the time. This data can be used to manipulate your bracket choices. This isn't the only way to mathematically simulate the games. Some use a point spread given by Vegas betting odds and convert these point spreads into winning percentage. The Bradley-Terry model converts computer rankings into probabilities. Of course, no method can perfectly account  for the twists and turns that occur in the tournament. Every year there are upsets that result in ruined brackets and one lucky guy who decided to take a chance on the underdog. The favorite this year is Kansas, and CBS Sports calculated the chance of a Kansas victory to be 16%.



Baseball

    Busy swinging the bat and keeping their eye on the ball, baseball players aren't consciously thinking about the mathematics that make their sp ort possible. However, hitting a home run or even making it to first base is centered heavily on angles, velocity, and energy. The speed of a pitch or of a ball after it's hit can be found using a specific equation. Ho is the height from which the ball is thrown, α is the angle at which the ball is thrown, vo is the speed at which the ball is thrown, and x is the distance that the ball travels. From the graph below, it can be seen that hitting the ball at a 45 degree angle will cause the ball to travel the farthest. 
Distance baseball will travel
Graph showing range with different α's.
Black graph 
α = 30o, blue graph when α = 45o, red graph when α = 60o

Projectile motion of a baseball
It can be argued that any sport involves similar physics. So why was baseball one of the sports chosen to examine with a mathematical lens? Mathematical approaches to managing baseball teams have surfaced throughout the sport. In fact, the term sabermetrics specifically describes the way in which statistical analysis is applied to baseball records. The term was coined from the acronym SABR (Society for American Baseball Research), and is used to evaluate and compare baseball players. Sabermetrics takes an emotionless, objective approach to baseball. It aims to answer only questions that can be proven with facts. Perhaps one of the most celebrated proponent of sabermetrics is Billy Beane, who inspired the movie Moneyball. Beane was the General Manager of the Oakland Athletics and used statistical analysis to lead the A's to a winning season. He looked at the risk factor of each player he brought into the program, and examined who was worth the money. He looked for players who may not have carried a famous name, but could still contribute to the team. In other words, he stretched the dollar and looked for economically smart choices. This is one example of how sabermetrics and analyzing stats is prominent in the sport of baseball.

Soccer

     Soccer is the world's most popular sport, and every day millions of people around the world take the field– from professionals to small children. However, I doubt any of these players take the time to examine the geometry that makes a soccer team successful. The basic shape of soccer is a triangle. the players on the field are connected by imaginary triangles, that build upon each other to diamonds and other shapes. Why a triangle? It allows for the most passing lanes and provides an option in every direction. The best teams in the world are known for being able to move the ball around the field in these triangles. 
     Free kicks provide another use for the application of geometry. Defenders line up in a wall, in hopes to impeded a direct path to the goal. The wall is set up at a specific angle and the person kicking the ball tries to bend the ball behind the ball. However, the amount of people that stand in the wall is dependent on where around the goal the ball lies. While the goalie isn't thinking about math when he/she sets up the wall, the logic behind it is definitely mathematical. The chart below shows that as the ball moves away from the center of the goal, less people are needed to defend the wall. When the ball is in the middle of the goal, the shooter has a larger angle, so the width of the wall needs to be greater. 




















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