the ramaz mathematical publication
TRANSCRIPT
Xevex Fall 2020
The Ramaz Mathematical Publication
Editors Akiva Shlomovich ‘21 Sophia Rein ‘21 Rachel Freilich ‘22 Eric Kalimi ‘22
Contributors Benjamin Meyer Yazdi ‘22 Daniel Kalimi ‘23 Ariella Golobrodsky ‘24 Jenny Davis ‘23 Finley Horowitz ‘22 Spencer Rubenstein ‘21 Rebecca Kalimi ‘23
Table of Contents
1
The Math of Auctions Daniel Kalimi
… 2
Risk and Die-Rolling Probabilities Benjamin Meyer Yazdi
… 3
The Math Behind Counting Cards in BlackJack Ariella Golobrodsky
… 4
The Math and Chemistry Behind Curly Hair Sophia Rein
… 6
So Many M&Ms! Rachel Freilich
… 8
The Physics of a Jump Shot Jenny Davis
… 8
The Mathematics of Winning the Lottery Rachel Freilich
… 9
How Many Chess Games are Possible? Daniel Kalimi
… 10
Terrence Tao Akiva Shlomovich
… 11
Numbers and Language Finley Horowitz
… 12
Blaise Pascal Daniel Kalimi
… 13
Probability, Expected Value, and the Argument to Move the 3-Point Line Spencer Rubenstein
… 14
Artwork Rebecca Kalimi
… 15
2
The Math of Auctions Daniel Kalimi
The earliest form of auction was in
Ancient Greece, 500 BC where men auctioned women to other people to marry. Later, in the Roman Empire, it became common for people to use auctions to liquidate property. In America, people used auctions in the south to sell slaves. Auctions have since evolved to more exclusively sell art or jewelry and to encourage people to donate more money at charitable events.
In the world of mathematics, when
discussing auctions, the name John Nash usually springs to mind. He is famous for coming up with the Nash equilibrium which states that in any competitive situation where both participants are rational, and both know that the other is rational, then there is always a best strategy. A commonly used example to explain the Nash equilibrium is the prisoner dilemma. In this famous example, two prisoners who can’t communicate are told that they can either betray the other person by testifying against them or remain quiet. If they both betray each other, they both get 3 years in prison, if they both remain silent, they both get 1 year in prison, and if one betrays, and the other is
silent then the one who betrayed gets set free and the other one gets 5 years in prison. The Nash equilibrium in this example is for both players to betray each other. Even though remaining silent leads to a better outcome if one prisoner is silent and the other betrays, one prisoner's outcome is worse.
Another mathematician, Thomas
Palfrey, used the Nash equilibrium to study the math of auctions. At the time, there was an argument over the reason for overbidding; some said that it was because people valued the satisfaction of winning, and others said that it was because people would rather overbid than lose an item. Palfrey did an experiment where he held an auction and he told the people participating the value of every item. Each person would get only one bet, and if they ended up winning the item at a lower price than its assigned value, they get to keep the difference. In this experiment, most people overbid, which proved that the primary reason for overbidding was the fear that they would lose the item to another person. In every auction, there is the perfect balance between making a profit and losing the item to another bidder.
3
Risk and Die-Rolling Probabilities Benjamin Meyer Yazdi
Risk, a strategic world-domination
board game, is known and beloved by many. The goal of the game is to conquer all countries on the map, which is done by invading adjacent, enemy owned countries.
The livelihood of one's soldiers is determined by rolling six-sided dice. Each die represents a single soldier’s battle performance, which may lead to its own death or the death of an enemy soldier.
The defender can roll two dice for any given battle round unless there's only one troop in that country, in which case only one die is rolled. The attacker may roll the number of troops attacking, which is minus one of the troops in the attacking country, with a maximum of three dice per round allowed.
The players engaged in battle roll their respective amount of dice and then match the highest and second-highest (if applicable) rolls against each other. For every match where the attacker’s dice show a higher number, the attacker can take one of the opponent’s soldiers engaged in battle off the board (indicating its death). If the paired dice show the defenders, the defender may do the same to the attacker.
In playing the game, much strategy is derived from the probability of winning a battle. This can help decide how many troops to fortify countries with, or whether to attack an opponent. To do this, most people use empirical data, which often leads
to a surprising loss. However, the days of guessing the odds are over! Thanks to Jason A. Osbourne at North Carolina State University, we now have a definitive solution and know the probability of winning a battle for different numbers of attacking and defending troops.
By introducing the Markov Chain and transition probability matrices, Osbourne was able to mathematically find the probability of an attacker winning a battle for different numbers of troops on each side of the battle-field, as shown in the table below. Note that in the table, “A” stands for attacker, and “D” stands for defender.
As you can see, the attacker has a
considerably better chance of winning a given battle than the defender. Most notably, when both players engaged in battle have an equal number of troops greater than five, the figure shows the probability of the attacker winning is more than 50%. Logically speaking, this means that the attacker should be more aggressive than one might originally assume. Now, the next time you play Risk with your family or friends, make sure to use this information and win!
4
The Math Behind Counting Cards in BlackJack
Ariella Golobrodsky
“People, please. We had a total of 76 cards that came out of the deck. Twenty-three were high cards with the value of minus one. Seventeen were neutral with no value at all, and the rest were low cards with the value of plus one. How could you lose the count?” exclaimed Mickey Rosa from the famous Hollywood debut, “21.” 21 is a movie based on a true story where five MIT undergraduates devise a system using card counting to beat all major casinos in LA at blackjack to make over 3 million dollars in one year. After watching this movie about the art of counting cards in blackjack, the viewer is left with the impression that counting cards is an ability only for the mathematically talented. Well, truthfully, it is not.
Counting cards is a method used to ensure victory for the player in blackjack. Blackjack is a popular game usually played in casinos between one dealer, and one or more players. Each player’s goal is to beat the dealer. The game begins with the dealer giving himself and each player 2 cards. All cards are facing upwards except for one of the dealer’s cards. The player makes their first move; they have two choices: either ask for another card (“hit”) or not (“stand”). The goal of the game is to have cards that add up to, but do not go over, 21. If the player is dealt a 10 and an Ace from the start they got a blackjack and automatically
win the game. If the player’s cards add up to more than 21 or the dealer’s cards value closer to 21, then the dealer wins the game.
In Blackjack, low cards are good for the dealer because they keep him safe. The dealer’s goal is to wait for the player to reach over 21 and keep the value of their cards around 16 or 17 to stay safe. The player wants high cards because the player’s goal is to get high cards to increase their chances of beating the dealer. Obviously, the removal of aces from the deck hurts the player tremendously because he or she will no longer have a chance to get a Blackjack. But what about all the other cards in between? How are the player’s chances of winning affected by the removal of other cards?
In Peter Griffins, The Theory of Blackjack , he calculates the effect of the removal of cards in a one-deck game. His chart explains how when certain cards are dealt the player is at an advantage, and when other cards are dealt the player is at a disadvantage. Above, we see this chart that
5
tells us when low cards (2-6) are dealt out the player has an advantage, when cards 7 and 8 are dealt out they give the player a slight advantage, but when high cards (9-A) are dealt out the player is at a big disadvantage. Each card dealt out alters the player’s chances of winning by different amounts. How did he calculate these values?
This chart was created using probability. For example, let’s say one deck is being played. To figure out the probability of being dealt a certain card we use a probability formula. “ x ” represents any card, and “ nx ” the number of cards with the value x that were already dealt out to players or the dealer “ nv ” represents the total number of cards that were dealt out. The image below shows the probability formula for receiving a specific card while playing a game with one deck of cards
Imagine playing one on one with the
dealer and you have a Queen, 2, 4, and an Ace which adds up to 17, and the dealer’s face-up card is 4. To reach the value of 21, you are going to need a 4, what is the probability of getting that card?. x=4, because that is the desired card, nx=2 because there were 2 cards valued 4 that were dealt out, and nv=5 because there have been 5 face-up cards dealt out in total. Below is the formula, and the probability of getting a 4 on your next turn.
.
Using this probability formula, we can understand how one would be able to calculate the probability of getting a certain card during the game. We also know that ideally, the player wants to be dealt with high cards to get a blackjack, or to reach the value of 21. So, if the player wants a 9, 10, or 11, the more of those cards left in the deck, the higher his chances of winning, as seen above. The more of any card left in the deck the higher your chances of obtaining it. And, the more low-value cards dealt out, the more of a chance the player has to be dealt a high-value card. This is how Peter Griffin created the “effects of the removal” chart. For example, removing one 10 from the deck decreases the chances of getting 21 will be significantly lower, specifically .51% lower. This is because there is less of a probability to pick a high card, and eventually reach 21, without that 10 cards.
Of course, not everyone playing blackjack will be able to calculate or memorize these percentages, and therefore in 1963, mathematician Harvey Dubner created the Hi-Lo system to help a player decide if they should Hit or Stand. He grouped the cards into two groups: High cards and Low cards. Low cards range from 2-6 and were the cards that gave you a great advantage if dealt out. Cards 7 and 8 had a minimal positive effect on the game, as seen in the “Effects of removal” chart, so they were left out of the Low group. And High
6
cards, the cards of greatest value to the player, were 9-A and had a negative effect on the players game if dealt out. The system assigns Low cards with a +1 value when dealt out, and High cards with a -1 value when dealt out. And now, all you need to do is keep the count. The way you do this is by adding up all the cards that have been dealt so far and using the point value system. The higher your sum is the more of a chance you have of winning, an indication to Hit, and the lower your count is the more likely you’re going to lose. So counting cards is really not as complicated as it seems in the movie 21. Using the Hi-Lo system, anyone can do it, even if they can’t understand the math behind it! The only downside is that you may get caught and get banned from casinos, so be careful!
The Math & Chemistry Behind Curly Hair Sophia Rein
Have you ever looked in the mirror
at your frizzy curls and wondered how math could possibly explain the chaos? I have! Turns out that your unruly kinks can be dissected mathematically (and chemically) in the same way that we describe DNA helices
and other oscillating waves. Human hair is surprisingly complex,
composed of many elegant symmetries. On a microscopic level, hair is composed of a Keratin protein backbone, which is made up of long amino acid chains. Keratin, similar to most fibrous proteins, aligns itself in an alpha helix conformation (a-helix, for short). As you can see in the diagram below, in an a-helix conformation peptide bonds sit comfortably on the inside of the spiral chain. Hydrogen bonds — the orange circles — align themselves on the outside of the spiral, parallel to the axis of the helix.
Sulfur is scattered throughout the
amino acid chains in human hair. When two Keratin strands are adjacent to one another, the Sulfur molecules interact. Specifically, the -SH bonds in Cystine groups form disulfide bonds (S-S) between the two strands through oxidation, a chemical process in which electrons are given away. As shown in the diagram below, this chemical reaction ties the two Keratin strands together into a microscopic helical pattern. When enough hair bonds into helical formation, the pattern becomes macroscopic - visible to the naked eye. Generally, the more Sulfur there is in your hair’s amino acid chains, the
7
curlier your hair will be.
There are other factors aside from Sulfur levels that contribute to the degree of curliness in hair. For starters, the shape of hair follicles can have a large impact. Teardrop-shaped follicles produce much curlier hair than cylindrical follicles because the hair grows out of the scalp at a larger angle. Similarly, the diagram below shows that as the angle of emergence from the scalp increases, the strand of hair becomes curlier. The cross-sectional geometry of hair strands also plays a role in curliness. Cylindrical hair strands are typically straight, oval hair strands are usually wavy and a cross section shaped like a flattened oval will almost always result in tight kinks. The diagram below also outlines different cross sections and their resulting hair types. This occurs because the shape of the cross section will tilt the circumference of the hair strand, resulting in varying degrees of curls.
Like every three-dimensional helix
pattern in nature, the coils in hair can be described with vectors. Specifically, we can use the following parametric equations in three dimensions to describe a curl’s spiral: x(t) = a⋅cos(ωt), y(t) = a⋅sin(ωt), z(t) = t In this function, ‘a' is the radius of the helix, a measure of how far in each direction the helix reaches. The sine and cosine functions create a circle of radius 'a' and Omega (ω) is the angular frequency, a measure of the helix’s rotation rate. Specifically, the Omega refers to the angular displacement per unit time of change that the helix undergoes.
In the end, no amount of hair products or treatments can overpower your hair’s natural structure. But, I do have some beauty tips! If you are looking to tame your curls, I recommend allowing gravity to weigh down your hair. You can really make use of gravity’s hair care benefits by wetting your hair to increase its weight. On the flip side, if you would like a frizzy perm try placing positively charged magnets next to your hair. Your hair has a strong negative charge because of all the Sulfur, so it will be attracted to the positive magnets and jump towards them. With Chemistry and Math by your side, you’ll never have another bad hair day!
8
So Many M&Ms! Rachel Freilich
The Fermi Method is an estimation
technique, founded by Enrico Fermi, used to estimate absolutely anything. Scientists use the Fermi Method in conjunction with other techniques to roughly determine quantities.
An example of a fermi problem would be: How many packets of M&Ms are required to make a single line of M&Ms with a distance of 100 meters?
1. Each packet of M&Ms has 30 pieces of chocolate
2. Each M&M has a diameter of 1 cm 3. There are 100 cm in a meter 4. It would take 10,000 M&Ms to make
a single line with a distance of 100 meters
5. 333 packs of M&Ms would be required to make a single line with a distance of 100 meters
The Fermi Method of estimation can be used to solve ridiculous, crazy, and efficient problems.
wrist, releasing the ball. He scores, bringing the game into overtime. This action might look effortless, but it takes a lot of practice to make the perfect jump shot. One needs not only intense skill, but also a strong understanding of how the mechanics of a jump shot work.
Most basically, the jump shot has four parts. First, the player must plant his feet shoulder width apart with the dominant foot marginally in front of the other. Next, the player must bend his knees and lift his arms. Both elbows must start somewhat on the side of the body. The elbow of the arm which is guiding the ball should face directly at the basket the entire time. After getting into this position, the player must jump. Once he is in the highest point of the arc, he must release the ball with a swift flick of the wrist.
The mechanics of a jump shot requires physics, geometry and psychology. The levers used to activate the shot are a part of physics. The levers need the right amount of force to activate them as dictated by Newton’s First Law of Inertia. It states that every object will stay at rest unless an outside force causes it to change its state. The amount of potential energy created by the first three levers allows the last lever to convert into kinetic energy which then pushes the ball towards the hoop. Next is Geometry. In order for the ball to go in the right direction the player must align his body to the basket for the shot, precisely calculating angles and distances at every point until his release. Lastly is psychology. The player must have confidence in his ability
The Physics of a Jump Shot Jenny Davis
The score is 101-99. The Golden
State Warriors are down one basket with four seconds left; the championship is at stake. Suddenly Stephen Curry gets the ball and plants his feet shoulder width apart and his dominant foot slightly in front of the other. He then bends his knees, tucks his elbow into his body, jumps, and flicks his
9
to perform and he cannot hesitate. While it sounds like a lot to think about in the last four seconds of a game, it is crucial to understand and practice these steps for success. Practicing everyday like Stephen Curry eventually creates muscle memory that makes it as natural as counting 1, 2, 3.
using combinatorics, there is a way to improve your odds of winning. For example, it will be easier to win the lottery if there are less balls to be drawn. Therefore, there are less ways that a certain amount of balls can be chosen. Additionally, lotteries with a smaller pick size will also increase your chance of winning. A lottery that picks 8 balls will have less combinations than a lottery that picks 10 balls.
In order to calculate the odds of winning the lottery, we use the binomial coefficient formula:
Where n equals the number of balls and r equals the pick size.
Therefore, in a lottery system with 49 balls where you chose 6 balls, the formula is:
₉C₆ 3, 83, 16₄ = 49!6!(496)! = 1 9 8
In this lottery system there are 13,983,816 different combinations and one’s chance of winning is:
113,983,816
If you happen to win, you should know that you are not just "one in a million," but "one in 14 million.
The Mathematics of Winning the Lottery
Rachel Freilich
The lottery remains one of the most compelling games in the world. If you win, you are definitely “one in a million”. Since the lottery is a totally random game, only math can explain the strategies of winning the lottery.
The chance of winning the lottery is very slim. The odds of winning the grand prize of the US powerball is 1 in 292,201,338 and even more miniscule for the Mega Millions. Only one in 302.6 Million people win the Mega Millions. In order to win the US powerball, it would take 292 million tries. If you attempted 100 tickets a week, it would take 2,920,000 weeks, or 56,154 years. If one would follow this trend, they would lose their money very quickly. Suppose a lottery ticket, on average, costs two dollars. After only one week, you would have already spent two hundred dollars.
Mathematically, the only way to elevate your chances of winning is to buy more tickets. No machine or person can predict the winning number before it is revealed to the rest of the world. However,
10
How Many Chess Games are Possible?
Daniel Kalimi
A well-known story having to do with chess is the story of the person who created chess. Nobody knows the actual person who created the game, but there is a story in which the supposed creator showed the game to a king, and the king offered him any reward he wanted. A smart man, he requested only one grain of rice that doubled for every square on the chessboard (1 for the first, 2 for the second, 4 for the third…). The king agreed, thinking it was a small price to pay but a week later the treasurer told the king that once you got halfway through the board, there would be more grains of rice than there are in the entire kingdom. After going through the entire board, the king would have to owe the creator over 18 quintillion ( ) .8 0 1 · 1 19 grains of rice. Instead, he just killed the inventor for trying to fool the king. This is a story taught to people to help understand the power of exponential growth. This problem is just using a chessboard, but what about chess itself? How many possible games are there? Is it true that there are more possible chess games than atoms in the observable universe?
The first real estimate of how many possible games there are is from mathematician Claude Shannon. He wrote a paper in the 1950s called “How to Program a Computer to Play Chess” and in the paper he gives a quick estimate for the number. He came up with what we now call Shannon’s number, (1 with 120 zeros!). The 0 1 120 number of atoms in the observable universe is approximately , so according to 0 1 80 Shannon’s number you could assign billions of games of chess to every single atom. He made his calculation by saying there are about 30 moves for every position and games last about 80 plies so his calculation was just which is about . In chess 0 3 80 0 1 120 terminology, a ply is when one person goes whereas a move is when one person goes and then the other person goes. This is just an estimate, but the actual number is much higher. For the first ply there are 20 possible moves and for the second ply, black can respond with the same 20 moves so by ply 2 there are 400 possible games. At ply 3, the number shoots up to 8,902 games that can happen and at ply 4 there are 197,772 possible games that can happen. The number is growing at an insane rate but, how many plies can there be in a game? Technically, a game could go on forever if you keep moving your pieces back and forth so there are boundaries set where if it goes 50 moves without a piece getting captured or the board repeating 3 times then it automatically becomes a draw. Under these conditions, the longest chess game can go on for about 11,800 plies. Based on this, famous mathematician G. H. Hardy came up with an
11
estimate of . Shannon’s estimate was 10 1050 miniscule compared to Hardy’s estimate, but it is important to remember that Shannon’s estimate was more realistic for normal, everyday chess whereas most of the games in Hardy’s estimate are nonsense games that would never be played in real life. Still, if you would want to have an estimate of good game of chess, there are an average of 3 sensible moves per ply and with an average of 80 plies per game that is which is 380 around possible chess games which is 0 1 40 an insanely large number but still not as many atoms in the observable universe.
countries send six students to solve six incredibly challenging math problems for a total of 42 points. He competed in 1986, 1987 and 1989, winning a bronze, silver and gold medal respectively. He is the youngest person in the 60 years of the IMO’s existence to win a gold medal. He published his first paper at 15, and at 16, graduated with both his bachelor’s and master’s degrees from Flinders University. He then received a scholarship to do his doctoral research at Princeton University, and received his doctorate in mathematics at age 21. Keep in mind that most American college students graduate with their bachelor’s at 22! That same year he received his doctorate, he became a faculty member at UCLA, and three years later became a professor, at age 24. Once again, he was the youngest ever. Over time he has amassed an astonishing collection of awards such as the MacArthur Award. But most notably, in 2006 he won the Fields Medal, the “Nobel prize” for mathematics.
So what is Tao doing now? Not much, except for bringing us closer to solving one of the most difficult and intriguing conjectures of all time: the Twin Prime Conjecture. The Twin Prime Conjecture is a question in mathematics which asks if there are infinitely many twin primes. What are twin primes? 3 and 5, 11 and 13, 17 and 19 are each twin primes. Twin primes are two primes only separated by one number. So does this pattern continue forever? Will you always be able to find a twin prime?
Terrence Tao Akiva Shlomovich
Terrence Tao is a name almost every math enthusiast knows, or at least should know. Tao was born in Australia in 1975 to two immigrants from Hong Kong. Tao was a mathematical prodigy who at age nine began taking college level mathematics courses. At age ten he began competing in the International Mathematical Olympiad (IMO), a competition where more than 100
12
Tao, along with many other mathematicians created the Polymath Project, a project designed to solve hard math problems. One mathematician, Yitang Zhang, helped move the project of solving the Twin Prime Conjecture along tremendously when he found the first finite bound between primes. In 2013, he found that the biggest gap between two prime numbers was 70 million numbers. Using his method and optimizing it, Tao and his colleagues were able to bring the number down to six! This means that primes that have six numbers in between them occur infinitely many times. These breakthroughs take a lot of time and effort to come to, which is why this problem has been around for over 100 years. Maybe one day we will witness Tao solve this problem, or maybe you will!
numbers, such as eleven or twelve, are very illogical because their names don’t even acknowledge that they are two digit numbers. Many East Asian languages, such as Mandarin, Japanese, and Korean, are more logical. They express their two digit numbers as the tens digit, the word ten, and then the units digit (ex. 87 would be “eight ten seven”). On the other hand, languages such as French and Danish are much more confusing. In French, seventy is phrased as “sixty and ten”, eighty is “four twenties”, and ninety is “four twenties and ten”. In Danish, fifty is “half three twenties” (2.5 * 20), sixty is “three twenties”, seventy is “half four twenties”, and so on until ninety. These languages express certain numbers in a base-20 system, which is very illogical in a number system in base-10 for everything else. In Dutch, the ones digit is written before the tens digit, which is also illogical.
These different complexities in languages affect the speed at which math is done with these numbers. When given a list of numbers to look at quickly and memorize, most English speakers would only be able to remember seven digits, whereas a Chinese speaker would most likely be able to remember ten. This is because most English numbers take about a third of a second to recite in their head while most Chinese numbers take only about a fourth of a second. This might not seem like a big difference, but it does in a verbal memory loop, which people use to recite the numbers in their head while reading them. The verbal memory loop can store only two seconds worth of information. Scientists
Numbers and Language By Finley Horowitz
Math is a universal language, but
research shows that the way people perceive numbers is heavily affected by the language they speak. Languages vary in how logically they label numbers. English, for example, falls towards the middle of that range. It expresses numbers with the tens position first and adds a -y to the end of some variation of the number in the tens place (ex. Four becomes forty, two becomes twenty). This makes sense because it is phrased in the same order that the numbers are written but using the -y to indicate the tens digit is not very logical. Certain
13
have tested this in several languages and have concluded that there is a strong correlation between how long a number takes to recite and how many a person can remember. Other studies show that children who speak languages with more complex numbers take more time to process simple math questions, such as estimating the place of a number on a number line. Similar hypotheses have been tested on adults and even then, there is a slight processing delay. For example, Dutch speaking adults, who express the numbers with the ones digit first, usually glance at the inverse of a number before finding the correct one on a number line. It has also been found that people who speak different languages use different parts of their brains when remembering numbers or doing arithmetic. This information shows that numbers may not be as universal as they may seem.
God's existence are trivial. Consequently, if God is real, the reward would be infinite. On the flip side of this, if you are a bad person in this world, you get short-term happiness (fun in this life) but long-term suffering (in hell). Another creation of his was the first mechanical calculator. His father, Etienne Pascal, was a tax collector in France, so Pascal developed a mechanical calculator to help his dad with his calculations. The calculator is known as the Pascaline and never became widely produced and sold because it was impractical and expensive. Rather, it became a status symbol for wealthy people in Europe. For actual mathematics, he used probability theory, which is a revolutionary theory that is still applied to gambling and is important regarding economics. From this came expected values, a mathematical measure of the worth of making a specific gamble.
Blaise Pascal Daniel Kalimi
Blaise Pascal was a famous
mathematician, physicist, inventor, philosopher, writer and Catholic theologian born on June 19,1623 in Clermont-Ferrand, Auvergne, France. People mostly know him for the “Pascal’s wager”, a philosophical argument he came up with. “Pascal’s wager” claimed that it is mathematically worth it to believe in God. Since, there is an infinite reward in the afterlife if you are a righteous person, how much time and effort one attributes to God and the possibility of
This is also the base idea for Pascal’s wager. Pascal also studied hydrodynamics and hydrostatics, which led to his invention of the syringe and the hydraulic press. Overall, Blaise Pascal was an influential man, whose discoveries still play a major role in today’s society. He died on the 19th of August 1662 at the young age of 39, but still did so much for the world in such a short period of time.
14
Probability, Expected Value, and the Argument to Move the
3-Point Line Spencer Rubenstein
Probability theory is the branch of
mathematics that deals with the outcomes of random and unpredictable events. Of all the areas of mathematics, probability theory is one of the most practical and applicable areas of study. The rules and outcomes of probability play a role in common areas of life such as finance, medicine -- and sports.
A fundamental concept of probability theory is that of expected value. The expected value of a variable is equal to the sum of all possible values, each multiplied by the probability of its occurrence. For example, in a simple coin toss game in which the player receives $10 for a toss of heads and $0 for a toss of tails, the expected value of the payout would be $5 (calculated as [$10 * 50%] + [$0 * 50%]).
Now, how does this all relate to basketball? For the last forty years since the 3-point line was first adopted by the NBA (note: it was introduced into other national basketball leagues many years before), the expected value of a 2-point shot and a 3-point shot were roughly the same. The expected value of a 2-point shot was approximately 1 (2 * 50%), and the expected value of a 3-point shot was approximate 1 (3 * 33%). This led to an even and balanced game as players and teams would have to weigh the risk of shooting a longer-distance
shot with the potential reward of earning an additional point.
However, as players' skills and specialization have improved over the years, the probability of making a 3-point shot for many of the elite players has increased well above 33%, resulting of the expected value for 3-pointers to often exceed the expected value for 2-pointers. This has led to a major increase in the number of 3-point shots taken throughout a game, and this is a trend that has developed across many (if not most) teams in the NBA.
Based on this development, many have argued that the 3-point shot should be made more difficult by increasing the distance in order to decrease the probability of a made shot and thereby bring its expected value back in-line with the 2-point shot.
By Rebecca Kalimi
Thank you to everyone who contributed to the Fall 2020 issue of Xevex!!
If you are interested in contributing to Xevex, please reach out to: Akiva Shlomovich [email protected]
Sophia Rein [email protected] Rachel Freilich [email protected]
Eric Kalmi [email protected]
15