Xevex Fall 2020
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
The Ramaz Mathematical Publication
Table of Contents 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
1
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 The earliest form of auction was in both players to betray each other. Even Ancient Greece, 500 BC where men though remaining silent leads to a better auctioned women to other people to marry. outcome if one prisoner is silent and the Later, in the Roman Empire, it became other betrays, one prisoner's outcome is common for people to use auctions to worse. 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.
The Math of Auctions Daniel Kalimi
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
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.
2
to a surprising loss. However, the Risk and Die-Rolling Probabilities days of guessing the odds are over! Thanks Benjamin Meyer Yazdi
to Jason A. Osbourne at North Carolina State University, we now have a definitive Risk, a strategic world-domination board game, is known and beloved by many. solution and know the probability of winning The goal of the game is to conquer all a battle for different numbers of attacking countries on the map, which is done by and defending troops. By introducing the Markov Chain and invading adjacent, enemy owned countries. The livelihood of one's soldiers is transition probability matrices, Osbourne determined by rolling six-sided dice. Each was able to mathematically find the die represents a single soldier’s battle probability of an attacker winning a battle for performance, which may lead to its own different numbers of troops on each side of the battle-field, as shown in the table below. death or the death of an enemy soldier. The defender can roll two dice for Note that in the table, “A” stands for any given battle round unless there's only attacker, and “D” stands for defender. 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 As you can see, the attacker has a their respective amount of dice and then match the highest and second-highest (if considerably better chance of winning a applicable) rolls against each other. For every given battle than the defender. Most notably, match where the attacker’s dice show a when both players engaged in battle have an higher number, the attacker can take one of equal number of troops greater than five, the the opponent’s soldiers engaged in battle off figure shows the probability of the attacker the board (indicating its death). If the paired winning is more than 50%. Logically speaking, dice show the defenders, the defender may this means that the attacker should be more aggressive than one might originally assume. do the same to the attacker. In playing the game, much strategy is Now, the next time you play Risk with your derived from the probability of winning a family or friends, make sure to use this battle. This can help decide how many information and win! troops to fortify countries with, or whether to attack an opponent. To do this, most people use empirical data, which often leads
3
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 “People, please. We had a total of 76 dealer’s goal is to wait for the player to reach cards that came out of the deck. over 21 and keep the value of their cards Twenty-three were high cards with the value around 16 or 17 to stay safe. The player of minus one. Seventeen were neutral with wants high cards because the player’s goal is no value at all, and the rest were low cards to get high cards to increase their chances of with the value of plus one. How could you beating the dealer. Obviously, the removal of lose the count?” exclaimed Mickey Rosa aces from the deck hurts the player from the famous Hollywood debut, “21.” 21 tremendously because he or she will no is a movie based on a true story where five longer have a chance to get a Blackjack. But MIT undergraduates devise a system using what about all the other cards in between? card counting to beat all major casinos in LA How are the player’s chances of winning at blackjack to make over 3 million dollars in affected by the removal of other cards? 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 In Peter Griffins, The Theory of the dealer’s cards. The player makes their Blackjack, he calculates the effect of the first move; they have two choices: either ask removal of cards in a one-deck game. His for another card (“hit”) or not (“stand”). chart explains how when certain cards are The goal of the game is to have cards that dealt the player is at an advantage, and when add up to, but do not go over, 21. If the other cards are dealt the player is at a player is dealt a 10 and an Ace from the disadvantage. Above, we see this chart that start they got a blackjack and automatically
The Math Behind Counting Cards in BlackJack Ariella Golobrodsky
4
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
.
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 Imagine playing one on one with the blackjack will be able to calculate or dealer and you have a Queen, 2, 4, and an memorize these percentages, and therefore Ace which adds up to 17, and the dealer’s in 1963, mathematician Harvey Dubner face-up card is 4. To reach the value of 21, created the Hi-Lo system to help a player you are going to need a 4, what is the decide if they should Hit or Stand. He probability of getting that card?. x=4, grouped the cards into two groups: High because that is the desired card, nx=2 cards and Low cards. Low cards range from because there were 2 cards valued 4 that 2-6 and were the cards that gave you a great were dealt out, and nv=5 because there have advantage if dealt out. Cards 7 and 8 had a been 5 face-up cards dealt out in total. minimal positive effect on the game, as seen Below is the formula, and the probability of in the “Effects of removal” chart, so they getting a 4 on your next turn. were left out of the Low group. And High 5
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 6
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!
7
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.
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
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
8
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.
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,
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:
₄₉C ₆ = 6!(4949!6)! = 13, 983, 816
In this lottery system there are 13,983,816 different combinations and one’s chance of winning is: 1 13,983,816
If you happen to win, you should know that you are not just "one in a million," but "one in 14 million.
9
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 ( 1.8 1019 ) 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, 10120 (1 with 120 zeros!). The number of atoms in the observable universe is approximately 1080 , so according to 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 3080 which is about 10120 . In chess 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 10
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 around 1040 possible chess games which is doctorate in mathematics at age 21. Keep in an insanely large number but still not as mind that most American college students many atoms in the observable universe. 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 Terrence Tao solving one of the most difficult and Akiva Shlomovich intriguing conjectures of all time: the Twin Terrence Tao is a name almost every Prime Conjecture. The Twin Prime math enthusiast knows, or at least should Conjecture is a question in mathematics know. Tao was born in Australia in 1975 to which asks if there are infinitely many twin two immigrants from Hong Kong. Tao was a primes. What are twin primes? 3 and 5, 11 mathematical prodigy who at age nine began and 13, 17 and 19 are each twin primes. Twin taking college level mathematics courses. At primes are two primes only separated by age ten he began competing in the one number. So does this pattern continue International Mathematical Olympiad (IMO), forever? Will you always be able to find a a competition where more than 100 twin prime? 50
estimate of 1010 . Shannon’s estimate was 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 380 which is
11
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 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
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
12
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.
This is also the base idea for Pascal’s wager. Pascal also studied hydrodynamics and hydrostatics, which led to his invention of Blaise Pascal was a famous the syringe and the hydraulic press. Overall, mathematician, physicist, inventor, Blaise Pascal was an influential man, whose philosopher, writer and Catholic theologian discoveries still play a major role in today’s born on June 19,1623 in Clermont-Ferrand, society. He died on the 19th of August 1662 Auvergne, France. People mostly know him at the young age of 39, but still did so much for the “Pascal’s wager”, a philosophical for the world in such a short period of time. 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
Blaise Pascal Daniel Kalimi
13
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 Probability theory is the branch of above 33%, resulting of the expected value mathematics that deals with the outcomes for 3-pointers to often exceed the expected of random and unpredictable events. Of all value for 2-pointers. This has led to a major the areas of mathematics, probability theory increase in the number of 3-point shots is one of the most practical and applicable taken throughout a game, and this is a trend areas of study. The rules and outcomes of that has developed across many (if not most) probability play a role in common areas of teams in the NBA. life such as finance, medicine -- and sports. Based on this development, many A fundamental concept of probability have argued that the 3-point shot should be theory is that of expected value. The made more difficult by increasing the expected value of a variable is equal to the distance in order to decrease the probability sum of all possible values, each multiplied by of a made shot and thereby bring its the probability of its occurrence. For expected value back in-line with the 2-point example, in a simple coin toss game in which shot. 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
Probability, Expected Value, and the Argument to Move the 3-Point Line Spencer Rubenstein
14
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 shlomovicha@ramaz.org Sophia Rein reins@ramaz.org Rachel Freilich freilichr@ramaz.org Eric Kalmi kalimie@ramaz.org
15