Ramaz Xevex - Winter 2024

XEVEX Winter 2024

Sarah Silverman ‘24 Grace Kollander ‘25

The Ramaz Mathematics Publication

3.14159265358

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TABLE OF CONTENTS

Probability and Gambler’s Fallacy Tricks to Become a Human CalculaJeremy Feder ’24

...3 tor Sara Kleinhaus ’27 ...8 The Secret to the Perfect Slice Sylvie Raab ’26 ...3 Faulty Statistics: The Case of

Sally Clark Geometry Proof Walk through: Avi Flatto-Katz ’25 ...9 Squared Circle Grace Kollander ’25 Large Numbers ...4 Sara Kleinhaus ’27 ...10 The Mathematics Behind Beethoven’s Music: Caroline Kollander ’27 ...5 Transformations in Math Education: A Historical Overview and Present Reforms Morris Cohen ’26 ...5

Geometry In The High School Curriculum: Pointless or Useful?

Sarah Silverman ’24 ...11 A Brief Overview of Game Theory Avi Flatto-Katz ’25 ...11

Greek Math: The Perspective of Plato Raymond Ashkenazie ’24 ...7 Ranked Choice Voting and the Math Behind It Ezra Gonen ’27 ...7

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Probability and Gambler’s Fallacy

Jeremy Feder

After flipping heads on a coin three times in a row, your friend asks you to bet on the next flip. Using your statistics knowledge, you know that the odds of flipping heads on a coin four times in a row are 0.54, which is a 6.25% chance, so you confidently bet that the next flip will be tails.

While this may seem like a good bet, you just fell into the trap of the Gambler’s fallacy. The Gambler’s fallacy is the common misconception that previous events influence the outcome of future events. In reality, each coin flip is independent, and the probability of getting heads or tails remains 50% on each flip.

A false perception of the law of averages leads many individuals to the Gambler’s Fallacy. The law of averages asserts that over a certain period of time, the particular outcomes of an event will approach its probability. In the case of the coin flip, you might have believed that there is a greater likelihood that tails is flipped because that would more logically contribute to the 50% average of landing heads, which had been 100% through the first three trials. However, this assumption neglects the fact that the law of averages asserts that a “certain period of time” must occur, and each individual trial cannot be predicted by this law.

Casinos rely on both the law of averages and the Gambler’s Fallacy to profit from players. Using the law of averages, a given game with a 30% win rate will produce a profit. While a few players might win and profit, as more and more players participate, they will approach a losing rate of 70%, and the casino will profit. As a given player

loses 70% of the time and begins to lose money, logically, they should stop playing. However, due to the Gambler’s fallacy, the player has a false sense of confidence that their odds are increasing and their losing streak will soon end. Next time Ramaz has a meeting discouraging students from gambling, all they need to do is explain probability.

The Secret to the Perfect Slice Sylvie Raab Imagine the disappointment of biting into a seemingly delicious cake only to find it dry around the edges. This is a dessert disaster that most of us have experienced at some point. But in 1906, Francis Galton solved this conundrum with his unconventional cake-cutting revelation. This scientific method of cake cutting is strange and not what you might expect.

Traditionally, cakes are cut into wedges. But, if the remaining cake is put into the refrigerator, it dries out and the next time someone takes a slice, it is not as moist or delicious as it was the first time.

On December 20, 1906, Francis Galton, aiming to solve this dilemma, published an article titled “Cutting a Round Cake on Scientific Principle.” Galton called the ordinary cake cutting method “very faulty” and suggested that consumers instead cut the first slice of cake straight down the center. This way, the two remaining halves of cake can be put back together and retain the moisture of the cake. Next, a slice should be cut in the opposite direction so that the result is two smaller,still-moist 3

slices of cake. From that point forward, you continue to cut slices of cake from the center so that it is always possible to put it back together in a way that will not dry out the cake and always taste as fresh as the first slice.

So why has this revolutionary technique not caught on in more than 100 years? One potential problem with this method: as illustrated in the image above, the pieces of cake get progressively smaller which is not exactly ideal when serving a large party. It also might be difficult to extract the first long slice of cake from the middle without crumbling the rest. But in pursuit of a perfectly moist slice, some sacrifices may be worth making.

Geometry Proof Walk through: Squared Circle Grace Kollander In the geometry proof Squared Circle we are given the following: Quadrilateral ABCD is a square (which tells us that the four sides are of equal length and that all the interior angles are 90 degrees). Then they tell us that line FG is a perpendicular bisector of line BC. Lastly, arc AC is part of circle B. Given all that information we have to find the measure of angle BED. The trick is to realize that because we are working with a circle that means any line that goes between B and arc AC is going to be equal because they are all radii of the circle.

The first step is to draw segment EC. Now we can recognize the relationship between triangle EBG and trian4

gle ECG. Both share side EG, and have congruent sides (BG and GC due to the perpendicular bisector), and they both have 90 degree angles. We can then conclude that by SAS, triangle EBG and triangle ECG are congruent.

That tells us that line segment EC is equal to line segment EB because of congruency. Since line segment EB is a radius of circle B and so is line segment BC we can say that line segments EC=EB=BC. Now we see that triangle BEC is an equilateral triangle because all of the sides of the triangle are congruent. The fact that it is an equilateral triangle is very important for this proof. It tells us that all the angles are congruent; so angle BEC is 60 degrees.

We know that line segment EC is equal to the radius of the circle. Additionally, the shape ABCD is a square, so line segment DC must be equal to line segment BC because that is one of the square properties. We already proved that line segment EC is equal to line segment BC, so therefore by the transitive property line segments CD = BC = EC. This tells us that triangle DEC is an isosceles triangle.

By subtracting angle measure GCE which we know is 60 degrees from angle measure BCD which is a right angle we find that angle measure ECD is 30 degrees.

Now we have one out of the three angles in the isosceles triangle DEC. Since it is an isosceles triangle we know that the other two angles are equal so we can label them both as x. We can now set up an equation, x+x+30= 180 degrees (sum of angles in any triangle). The next step would be to combine like terms and isolate the variable giving us, 2x= 150. Finally, we get that x= 75.

Remember we are trying to find an angle measure BED and now we have two

angles that together make up that angle measure. Angle DEC (75 degrees) and angle BEC (60 degrees) adds up to angle measure BED (75 degrees +60 degrees = 135 degrees). Finally we can conclude that angle measure BED is equal to 135 degrees.

The Mathematics Behind Beethoven’s Music: Caroline Kollander There is math behind every piece of music composed. Ludvig Van Beethoven was a famous composer and pianist during the seventeen to eighteen hundreds. Beethoven became deaf at 28 years old. This meant that he could not hear his own music, making the mathematics behind his pieces much more important. It was the only way he could understand what the music he was writing sounded like.

A method of Beethoven’s was repetition and variation. The repetition made for a steady unified piece. The variation aspect added impressive and engaging aspects to the music.

In Beethoven’s harmonies cords would move from one to another symmetrically by a process called harmonic progressions. The ratio between the different frequencies of pitch made for a blended and smooth composure.

Numerical relationships were also important to Beethoven’s music. Musical themes were repeated a certain amount of times before introducing a new one. Sections were made to last for a particular number of beats.

In measure 50 of Beethoven’s song “Moonlight Sonata”, the first part has three D major notes with spaces between them called thirds, skipping the next note in the scale. Ultimately, Beethoven’s work has a lot more depth to it than just the sound of it. He actually had to incorporate mathematics to create and sense it.

Transformations in Math Education: A Historical Overview and Present Reforms Morris Cohen Mathematics has played a very important role in shaping humans throughout time. Math education has undergone significant changes over the past centuries. The beginning of math education can trace back to ancient civilizations, where math concepts were normally done through day to day applications. Ancient Egyptians, for example, used geometry for land surveying, while Babylonians developed sophisticated algorithms for solving mathematical problems. We know that they did math from the ancient finding that historians dug up. On these tablets we find early many mathematical concepts that we use, even today.

These tablets date back to 3000-2000 BCE. Unlike the 10-base system which we use today, they used a base 60 system. The number concept of “one” was depicted as with “V” symbol, and numbers up to 60 were written out using a combination of these “V”s and “<“s, the symbol for 10. For numbers above 60, a different system was used. The Babylonian number 3 3, for example, meant 3 × 60 + 3, or 183. The Greeks, notably Pythagoras and Euclid, laid the foundation for geometric principles, such as pythogras’ theory of the right triangle, setting the stage for centuries of mathematical advancements. In the medieval times, the math began to shift from practical applications to theoretical understanding. The mainstream use of Arabic numbers and the decimal system helped improve the common math notation. Algebraic methods began to emerge during the medieval period, influenced by works translated from Arabic scholars. Mathematicians like Al-Khwarizmi contributed to the 5

development of algebra, introducing concepts like solving quadratic equations and linear equations. Islamic scholars kept and expanded upon Greek mathematical knowledge, introducing algebra and advanced trigonometry. Education in Europe during the medieval period was closely tied to religious centers. Monasteries and catholic schools were important centers of learning, and they played a crucial role in preserving and transmitting knowledge, including mathematical ones. However, access to education was limited, typically only for the elites.

The 19th century saw many significant reforms in math education. The Industrial Revolution introduced the need for a skilled workforce. Pioneers like Pierre-Simon Laplace and Carl Friedrich Gauss contributed to the development of modern mathematical notation and introduced more challenging mathematical reasoning.

In the early stages of formal education, mathematics instruction was fixed in rote memorization and procedure. Students were taught to solve mathematical problems through repetitive practice, only using algorithms and formulas rather than understanding concepts. This approach was made to shape individuals to perform arithmetic but typically did not have a deeper understanding of the main principles.

In the 1950s and 1960s, the New Math movement emerged in response to concerns that the older traditional math education was not enough to prepare students for the quickly changing technological work field.

This New Math was set to introduce more abstract mathematical concepts earlier in the curriculum, emphasizing set theory, logic, and mathematical structure. One of the key parts of the New Math was the idea that students needed to understand 6

the foundational rules of mathematics rather than relying solely on strict memorization of procedures.

However, the movement faced criticism for being too vague and not having practical applications, leading to confusion among both students and teachers. While it might have had good intentions, the implementation was so terrible that it may more harm than good. In Why Johnny Can’t Add: The Failure of the New Math, a book detailing the flaws of New Math, the author, Morris Kline said that New Math “ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations, if one does not know the older ones.” In other words, it was a disaster. This led to another shift in the education of mathematics.

The National Council of Teachers of Mathematics (NCTM) became more wellknown in the 1980s. They approached math education by emphasizing problem-solving, critical thinking, and real-world applications. It helped prepare students for an increasingly difficult and technologically driven society, encouraging teaching methods that go beyond rote review. Together with this, the Common Core State Standards (CCSS) emerged in the 21st century, offering a standardized look on math education across the United States. The CCSS provides a detailed and direct standard for math education per grade. Although it was, better, than New Math, there were still many flaws with the system.

To list some: Teachers and students must make an extra effort to adapt to new teaching and learning methods.

The standards lack specificity.

There is an increased requirement for high-stakes testing.

States that have higher standards must now accept lower standards due

to the CCSS. Many of these reasons are why we do not use CCSS today in Ramaz.

Overall, the journey of math education highlights the human need for better education. From ancient applications to modern problem-solving, each era has left its mark. The story of math education is ongoing, and its chapters continue to unfold in the classrooms of today and tomorrow.

Greek Math: The Perspective of Plato Raymond Ashkenazie Greek mathematics studies patterns, structures, numbers, and measurements. It has a rich history intertwined with the intellectual evolution of ancient Greece. In its connection between Greek mathematics and philosophy, Plato stands as someone who used geometry as a gateway to understand the fabric of reality. Plato believed that the abstract world of mathematics was crucial to showing us the forms that shape our existence.

One of his most important philosophies was the Theory of Forms. He showed that the material world is a reflection of non-material forms. He believed that these forms represented the perfect essence of imaginative concepts like beauty, justice and mathematical entities. Plato believed that Geometry is the language to grasp these forms he discusses. In “The Republic,” he illustrates this belief through his allegories of the divided line and the allegory of the cave.

He did that through the storytelling in the Allegory of the Cave, where prisoners were chained to face the

wall to perceive the shadows that were cast by objects behind them.

Plato shows the journey of liberation from the cave’s shadows as a mirror of his vision of enlightenment through the study of mathematics, particularly geometry.

He shows the shadows on the cave wall symbolize the deceptive nature of the physical world. He shows that only through the study of geometry could we break free from the shadows and ascend to a realm of eternal truths. Geometry unlocks the shackles of ignorance and perceiving the world as it truly is. Through this, we are able to see the importance of studying math.

Ranked Choice Voting and the Math Behind It

Ezra Gonen

In the majority of US states, a voting system called “first-past-thepost” is used. This system operates in a very basic manner. The candidate with the most votes wins the election. When there are two candidates, this system functions efficiently, because one candidate will have at least 50% of the vote. However, when there are multiple candidates, sometimes, the candidate with the most votes, may not have received support from the majority of voters. To solve this issue, ranked-choice voting has been used. The system works very well, especially when multiple candidates are vying for a few spots.

Voters are given a ballot on which they can rank the candidates in order of how much they support them.

A basic example of ranked-choice voting is if you have 4 candidates, candidate A, candidate B, candidate C, and candidate D. All four candidates are competing for 2 spots. For example, candidate A has 550 votes, 7

candidate B has 137 votes, candidate C has 200 votes, and candidate D has 113 votes. To secure a majority, a candidate must have a number of votes represented by the equation Total Number of Votes2+1. In the case of 1,000 voters, a majority would be 501 votes. In our example, candidate A has a clear majority and is therefore given the first seat. For the second seat, the candidate with the least number of seats is eliminated.

Therefore, candidate D is not able to win a seat anymore. The votes given to candidate D are now given to the candidates selected on the subsequent ranking levels. This can be shown as VT=VTXVCD , where VT is votes transferred to candidate X, DTX is the fraction of votes from Candidate D to X, and VCD is Votes for Candidate D. This means that candidates B and C would receive a certain number of votes based on the 2nd choice of those who voted for candidate D originally. If 50 voters listed candidate D as option one, and candidate C as option 2, then candidate C would receive 50 votes. The same concept would apply to candidate B. If people voted for candidate A in their second choice, then you have to look at choice 3, but the same formula applies. You eventually keep going through the process of redistributing votes until all the votes have been distributed. Then you look for a candidate with a majority to receive the 2nd seat.

Ranked-choice voting allows for every voter to have their voice heard until every last seat has been taken by a candidate. This voting system is the most fair because people no longer have to pick a certain candidate and instead can pick multiple candidates with whom they agree. Overall, ranked-choice voting is more effective than first-past-the-post voting and more mathematically sound.

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Tricks to Become a human calculator Sara Kleinhaus Although we all have calculators on our phones, it can be very useful to be able to do arithmetic in your head. There are a few tricks that we can use to operate on large numbers quickly. When multiplying two numbers close to 100 there is a pretty simple way that we can figure out these multiplication problems without doing very much work.

1) For example, we can use 91x98.

For the first step, subtract both numbers from 100, giving us 10098=2, and 100-91=9 2) Next, we subtract each number by the difference of the other, giving us 98-9=89, and 91-9=89 3) Next, we take the subtrahend of our last equations and multiply them, giving us 9x2=18 4) We can now take the difference, which is always going to be equal, which in this case is 89 5) The product of the two original numbers will be the answer to step number 4 in the thousands place, and the answer to 3 in the tens place, making the answer 8918 Here is the proof as to why this is trueLet x=(100-a) Let y=(100-b) xy=(100-a)(100-b) xy=100^2-100a-100b+ab xy=100(100-a-b)+ab Note that in any number abcd, the ab really represents 100*ab. Therefore, we can string together the (100-a-b) and ab to make our number, (100-a-b)ab We can also square numbers that end with 5 very easily. When we have n^2, the digit in the tens place

being 5, the number is always going to end in 25. We can figure out the numbers before by doing n(n+1). The answer to that would go before the 25, which would be the final answer.

Here is the proff as to why this works Because x ends in a 5, we can write x as (10a+5) x^2=(10a+5)(10a+5)=100a^2+100a+25=1 00(a(a+1))+25 Another cool mental math technique is that we can figure out very easily if a 3 or more digit number is divisible by 8. For example, in the number 4556634064, the last three digits are 064, which is divisible by 8, making the entire number divisible by 8.

Although these tricks only work in specific cases, there is no overall trick for mental math. However, some of the tricks that I have shown can be useful in high school math. Multiplying is something that always comes back, and so is squaring and square rooting.

Although you may not need to multiply numbers close to 100 very often, it is still pretty cool to be able to do those problems in your head very quickly.

Faulty Statistics: The Case of Sally Clark Avi Flatto-Katz One day, in December of 1996, Sally Clark found her 2 ½ month old baby, Christopher, unresponsive.

She brought him to the hospital, where he was quickly declared dead.

The coroner noted the cause of death as SIDS: Sudden Infant Death Syndrome. Just over a year later, in January of 1998, Sally’s second baby, Harry, died in the same circumstances. He was just 2 months old. In Harry’s case, the coroner

found recent bleeding at the back of the eyes and in the spinal cord, but he was otherwise healthy. Soon after Harry’s death, Sally was arrested, and accused of smothering both her children to death. Aside from the death of both her babies, there was no real evidence that she had smothered her kids. Most of the prosecution’s evidence came from the testimony of medical experts, who testified about the probability of both of her children dying from SIDS. In 1990, the risk of a baby dying of SIDS in the UK was 1 in 8,500, in middle-class families with no known risk factors. Assuming that two deaths from SIDS are independent events (which is very unlikely), the risk of two babies dying in a middle class family with no known risk factor would be the square of that: 1 in 2.75 million.

Following this, two deaths from SIDS in the same family would occur once every 7 years in England.

While very unlikely, it could definitely happen. However, one of the pediatricians who testified in the case, Sir Roy Meadows, made a dreadful mistake. He stipulated that since Sally was wealthier than the average middle class family, the odds of two of her babies dying from SIDS were much lower: 1 in 73 million. Sir Meadows was not a statistician, and had no authority or knowledge to make that stipulation. According to Sir Meadows’ statistics, two deaths from SIDS in the same family would only happen once every 185 years. This made Sally look much more likely to be guilty than what was actually the case.

In addition to making up statistics, Sir Meadows misrepresent-

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ed what those statistics meant.

He presented the 1 in 73 million chance of both of Sally’s babies dying of SIDS to be a 1 in 73 million chance that Sally was innocent. This was not the case at all.

The odds of two infants being murdered in the same household is just 1 in 2 billion. When comparing the chances that both her babies died of SIDS to the chances that both of her babies were murdered, it is then much more likely that her babies died of SIDS. It is therefore in fact much more likely that Sally was innocent.

Sally Clark’s case remains one of the most infamous examples of the misuse of statistics in court.

Sally spent 4 years in prison, and was only acquitted after medical evidence was discovered of a possible bacterial infection in Harry. One of the main takeaways from Sally’s case is that only statisticians should be allowed to testify about statistics. Only statisticians truly understand the statistics that are being presented, and would be able to explain the nuanced complexities that they come with. In Sally’s case, the medical examiners grossly misrepresented the statistics. While it is impossible to know exactly what the outcome would have been, it is much more likely that Sally would have been acquitted without this misleading testimony.

Large Numbers Sara Kleinhaus When you think of one million, it seems like a pretty big number, but what is the value of one million? A comparison that we can make is that if you were to stack 10

one million sheets of paper on top of each other, the stack would be over a hundred yards high - or the length of a football field. Considering the thinness of a sheet of paper, that is pretty tall.

If a million is manageable, a billion can’t be that much bigger right? Well, it is much bigger. It would take over a hundred years straight to count to a billion, and only 2 weeks to count to a million. But how much bigger does it get? Well, the short answer is, much much bigger.. Let’s start with one trillion. In order to count to a trillion, it would take 31,700 years. One hundred trillion has 14 zeros after the 1, and if you were to stack 100,000,000,000,000 dollar bills on top of each other, the stack would go to the moon and back 14 times. That’s a crazy amount of space.

Now if you think a trillion is massive, that’s nothing compared to a googol, also known as Graham’s number. A googol is 10,​ 000,​ 000,​ 000,​000,​000,​000,​000,​000,​000,​000,​ 000,​000,​000,​000,​000,​000,​000,​000,​ 000,​000,​000,​000,​000,​000,​000,​000,​ 000,​000,​000,​000,​000,​000,​000, which can much more easily be written as 10^100. To put this into perspective, it would take 317 novemvigintillion (1 with 90 zeros) years to count to googol. That’s approximately 10,​000,​000,​000,​000,​000,​000,​ 000,​000,​000,​000,​000,​000,​000,​000,​ 000,​000,​000,​000,​000,​000,​000,​000,​000 times the age of the universe. Another way to think about it is that there are 10^82 atoms in the known universe. The universe is humongous, and there are not even googol atoms in it.

While googol is absolutely massive,

it is minuscule compared to googolplex. Googolplex is one with a googol number of zeros after it. In order to even write out the number googolplex, it would take 10^92 years, which is 10^82 times the age of the universe. And that’s just to write out the number. The truth is that numbers continue to go on and on forever and ever, and the numbers that seem very big to us are actually nothing compared to how large numbers truly get.

Geometry In The High School Curriculum: Pointless or Useful?

Sarah Silverman Why do high schoolers learn Euclidean geometry? Ask around the Ramaz building, and a good number of 9th graders may be contemplating this exact question. The concepts of parallel lines and congruent angles seem divorced from the graphing, quadratic factoring, and mathematical buildup that comes later.

Most, if not all, students forget the minute details by senior year, leaving many left wondering what the point all was.

Besides being required educational material to graduate in New York State, Geometry enables students at the beginning of their high school career to grasp how math is not just the “plug and chug” system but rather can be conceptually understood. In fact, Geometry can only be conceptually understood since very few equations/ actual numbers are used in proofs.

For the architects in the classroom, Geometry will show up on their college curriculum, but for the rest of the student body, it serves as the most applicable math

to real-world problems. Hanging a picture frame? Trying to figure out if the volumes of two boxes you bought for packing up antiques are equal? That is Geometry!

Based in Greek texts, the history of Geometry is vast and interwoven with the development of the modern world. Besides basic arithmetic, which has stood as a useful guidepost since early times, Geometry is the oldest math we come across in our 1-12 education. But students have only been learning the concept in the United States school system since 1844, so to put that in perspective, not many students, taking in the scope of history, have sat in the same seat as you.

By taking geometry in high school, students learn math applicable to their everyday lives while growing their appreciation of the concepts behind the numbers. And who knows, maybe they might actually enjoy it!

A Brief Overview of Game Theory Avi Flatto-Katz What is it?

Game Theory is a branch of mathematics that studies decision making in different situations. It analyzes how rational decision makers-‘players’, make decisions in different scenarios, or ‘games’.

The different players have different ‘strategies’, or different options of acting in the given situation. For each player, the outcomes can be ranked in order of preference.

What is it used for?

Game theory can be applied in many different topics. It is used in economics to study pricing strat11

egies, auctions, and competition between firms. Computer Science utilizes game theory to create algorithms, especially for artificial intelligence. Additionally, game theory has been used in the study of evolution, by explaining how interactions influence evolutionary outcomes. It is also used in psychology, war, politics, and environmental science.

A famous example of game theory being used in the real world was during the Cuban Missile Crisis of 1962. Many of President John F Kennedy’s decisions at the time were influenced by his advisor Thomas Schelling, a notable game theorist.

Thomas Schelling described the crisis as a “competition in risk-taking”, and compared it to the game of Chicken(a model of conflict for two players in game theory).

History: Game theory was created in 1928, with the publication of John von Neumann’s paper on ‘The Theory of Games and Strategy’. Neumann was a noted mathematician, physicist, and economist, and had worked on the Manhattan project during the second world war. 16 years after publishing his paper, Neumann wrote a book on game theory titled:​​Theory of Games and Economic Behavior. Neumann later came to be known as the father of game theory. The study of game theory greatly increased during the 1950’s, during which John Nash discovered the idea of Nash Equilibrium, a key component of game theory. Game theory was first explicitly applied to evolution during the 1970’s. In 1994, the first nobel prize in game theory was awarded to John Harsanyi, John Nash, and Reinhard Selten. From 1994 through 2020, there have been

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16 Nobel Prize winners for game theory; one in relation to evolution, and 15 in relation economics.

Example problem: One of the most famous problems in game theory is the Prisoner’s Dilemma. It goes like this: Police suspect two people, A and B, of a crime. They have no evidence that either A or B committed the crime, other than that they were both at the scene of crime when it happened.

The police arrest both A and B. A and B are interrogated separately, and each one has no knowledge of the interrogation of the other.

At the end of the interrogation, A and B are each given 2 options: Confess, or remain silent.(Again, both A and B have no knowledge of the other’s decision) If both A and B remain silent, they each get 1 year in prison. If they both confess, they each get 5 years in prison. If one confesses and the other remains silent, the one who confesses goes free, and the one who remains silent gets 20 years in prison. All these possibilities are illustrated in the adjacent image: What is interesting about the Prisoner’s Dilemma, is that the best strategy for each individual is to confess, even though both would be better off if they both remained silent. The reason for this is because the potential loss of remaining silent(20 years), is much greater than the potential gain(only 1 year). Additionally, since A and B are not allowed to talk and corporate, there is no guarantee that if one remains silent, the other will too, and so there is a greater risk of getting the worst outcome(20 years).

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44431858676975145661406800700237877659134401712749470420562230538994561314071127000407854 73326993908145466464588079727082668306343285878569830523580893306575740679545716377525420 21149557615814002501262285941302164715509792592309907965473761255176567513575178296664547 79174501129961489030463994713296210734043751895735961458901938971311179042978285647503203 19869151402870808599048010941214722131794764777262241425485454033215718530614228813758504 30633217518297986622371721591607716692547487389866549494501146540628433663937900397692656 72146385306736096571209180763832716641627488880078692560290228472104031721186082041900042 29661711963779213375751149595015660496318629472654736425230817703675159067350235072835405 67040386743513622224771589150495309844489333096340878076932599397805419341447377441842631 Reach out to: 29860809988868741326047215695162396586457302163159819319516735381297416772947867242292465 Sarah Silverman (silvermans@ramaz.org) 43668009806769282382806899640048243540370141631496589794092432378969070697794223625082216 88957383798623001593776471651228935786015881617557829735233446042815126272037343146531977 Grace Kollander(kollanderg@ramaz.org) 77416031990665541876397929334419521541341899485444734567383162499341913181480927777103863 87734317720754565453220777092120190516609628049092636019759882816133231666365286193266863 3606273567630354477w628035045077723554710585954870279081435624014517180624643626794561275 Faculty Advisor: Dr. Fabio Nironi (nironif@ramaz.org) 31813407833033625423278394497538243720583531147711992606381334677687969597030983391307710 98704085913374641442822772634659470474587847787201927715280731767907707157213444730605700 73349243693113835049316312840425121925651798069411352801314701304781643788518529092854520 11658393419656213491434159562586586557055269049652098580338507224264829397285847831630577 77560688876446248246857926039535277348030480290058760758251047470916439613626760449256274 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