“Despite the widespread popularity of coin flipping, few people pause to reflect on the notion that the outcome of a coin flip is anything but random: a coin flip obeys the laws of Newtonian physics in a relatively transparent manner,” the. Diaconis has even trained himself to flip a coin and make it come up heads 10 out of 10 times. The outcome of coin flipping has been studied by the mathematician and former magician Persi Diaconis and his collaborators. If that state of knowledge is that You’re using Persi Diaconis’ perfect coin flipper machine. Julia Galef mentioned “meta-uncertainty,” and how to characterize the difference between a 50% credence about a coin flip coming up heads, vs. Bayesian statistics (/ ˈ b eɪ z i ən / BAY-zee-ən or / ˈ b eɪ ʒ ən / BAY-zhən) is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a degree of belief in an event. , Diaconis, P. This gives closed form Persi Diaconis’s unlikely scholarly career in mathematics began with a disappearing act. Read More View Book Add to Cart. D. We show that vigorously flipped coins tend to come up the same. Persi Diaconis was born in New York on January 31, 1945. Persi Warren Diaconis is an American mathematician of Greek descent and former professional magician. 5. Diaconis' model proposed that there was a "wobble" and a slight off-axis tilt that occurs when humans flip coins with their thumb, Bartos said. D. Amer Math Monthly 123(6):542-573. Diaconis' model proposed that there was a "wobble" and a slight off-axis tilt that occurs when humans flip coins with their thumb, Bartos said. In experiments, the researchers were. Persi Diaconis is the Mary V. 8. He received a. Persi Diaconis and Brian Skyrms begin with Gerolamo Cardano, a sixteenth-century physician, mathematician, and professional gambler who helped. 338 PERSI DIACONIS AND JOSEPH B. 51. A recent article follows his unlikely. Mazur Persi Diaconis is a pal of mine. We give fairly sharp estimates of. The probability of a coin landing either heads or tails is supposedly 50/50. The new team recruited 48 people to flip 350,757 coins. American Mathematical Society 2023. Persi Diaconis. A well tossed coin should be close to fair - weighted or not - but in fact still exhibit small but exploitable bias, especially if the person exploiting it is. An early MacArthur winner, he is a member of the American Academy of Arts and Sciences, the U. SIAM Rev. Persi Diaconis (1945-present) Diaconis’s Life o Born January 31, 1945 in New York City o His parents were professional musicians o HeIMS, Beachwood, Ohio. Sort by citations Sort by year Sort by title. com: Simple web app to flip a virtual coin; Leads in Coin Tossing （页面存档备份，存于互联网档案馆） by Fiona Maclachlan, The Wolfram Demonstrations. This challenges the general assumption that coin tosses result in a perfect 50/50 outcome. Apparently the device could be adjusted to flip either heads or tails repeatedly. As they note in their published results, "Dynamical Bias in the Coin Toss," the laws of mechanics govern coin flips, meaning that "their flight is determined by their initial. Stanford University professor, Persi Diaconis, has demonstrated that a coin will land with the same pre-flip face up 51% of the time. Frantisek Bartos, a psychological methods PhD candidate at the University of Amsterdam, led a pre-print study published on arXiv that built off the 2007 paper from. Suppose. At each round a pair of players is chosen (uniformly at random) and a fair coin flip is made resulting in the transfer of one unit between these two players. The trio. If it comes up heads more often than tails, he’ll pay you $20. He found, then, that the outcome of a coin flip was much closer to 51/49 — with a bias toward whichever side was face-up at the time of the flip. We call such a flip a "total cheat coin," because it always comes up the way it started. For people committed to choosing either heads or tails. Although the mechanical shuffling action appeared random, the. The Edge. I have a fuller description in the talk I gave in Phoenix earlier this year. With C. PERSI DIACONIS AND SVANTE JANSON Abstract. An analysis of their results supports a theory from 2007 proposed by mathematician Persi Diaconis, stating the side facing up when you flip the coin is the side more likely to be facing up when it lands. The outcome of coin flipping has been studied by Persi Diaconis and his collaborators. The historical origin of coin flipping is the interpretation of a chance outcome as the expression of divine will. Gambler's Ruin and the ICM. Diaconis and colleagues estimated that the degree of the same-side bias is small (~1%), which could still result in observations mostly consistent with our limited coin-flipping experience. Frantisek Bartos, of the University of Amsterdam in the Netherlands, said that the work was inspired by 2007 research led by Stanford University mathematician Persi Diaconis who is also a former magician. Forget 50/50, Coin Tosses Have a Biasdarkmatterphotography - Getty Images. Magical Mathematics reveals the secrets of fun-to-perform card tricks—and the profound mathematical ideas behind them—that will astound even the most accomplished magician. A most unusual book by Persi Diaconis and Ron Graham has recently appeared, titled Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks. This project aims to compare Diaconis's and the fair coin flip hypothesis experimentally. In this lecture Persi Diaconis will take a look at some of our most primitive images of chance - flipping a coin, rolling a roulette wheel and shuffling cards - and via a little bit of mathematics (and a smidgen of physics) show that sometimes things are not very random at all. Diaconis` model proposed that there was a `wobble` and a slight off-axis tilt that occurs when humans flip coins with their thumb,. When you flip a coin to decide an issue, you assume that the coin will not land on its side and, perhaps less consciously, that the coin is flipped end over end. 23 According to Stanford mathematics and statistics professor Persi Diaconis, the probability a flipped coin that starts out heads up will also land heads up is 51%. Find many great new & used options and get the best deals for Ten Great Ideas about Chance by Brian Skyrms and Persi Diaconis (2017, Hardcover) at the best online prices at eBay! Free shipping for many products!. Before joining the faculty at Stanford University, he was a professor of mathematics at both Harvard University and Cornell University. Professor Persi Diaconis Harnessing Chance; Date. The results found that a coin is 50. Dynamical Bias in the Coin Toss. Diaconis, P. The limiting chance of coming up this way depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Kick-off. October 10, 2023 at 1:52 PM · 3 min read. October 10, 2023 at 1:52 PM · 3 min read. Bio: Persi Diaconis is a mathematician and former professional magician. ダイアコニスは、コイン投げやカードのシャッフルなどのような. We analyze the natural process of flipping a coin which is caught in the hand. The limiting chance of coming up this way depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. AFP Coin tosses are not 50/50: researchers find a. An interview of Persi Diaconis, Newsletter of Institute for Mathematical Sciences, NUS (2) (2003), 12-15. Ten Great Ideas about Chance Persi Diaconis and Brian Skyrms. That means that if a coin is tossed with its heads facing up, it will land the same way 51 out of 100 times . You put this information in the One Proportion applet and. Consider first a coin starting heads up and hit exactly in the center so it goes up without turning like a spinning pizza. The autobiography of the beloved writer who inspired a generation to study math and. Persi Diaconis. It is a familiar problem: Any. Ask my old advisor Persi Diaconis to flip a quarter. According to the standard. 3. Persi Diaconis and Brian Skyrms. In a preregistered study we collected 350,757 coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by. The team conducted experiments designed to test the randomness of coin. , Ful man, J. Our data provide compelling statistical support for D-H-M physics model of coin tossing. Besides sending it somersaulting end-over-end, most people impart a slight. 8 per cent, Dr Bartos said. 2. Persi Diaconis, Mary V. "Gambler’s Ruin and the ICM. From. Second is the physics of the roll. This same-side bias was first predicted in a physics model by scientist Persi Diaconis. He could draw on his skills to demonstrate that you have two left feet. "Q&A: The mathemagician by Jascha Hoffman for Nature; The Magical Mind of Persi Diaconis by Jeffrey Young for The Chronicle of Higher Education; Lifelong debunker takes on arbiter of neutral choices: Magician-turned-mathematician uncovers bias in a flip of the coin by Esther Landhuis for Stanford ReportPersi Diaconis. Persi Diaconis Abstract The use of simulation for high dimensional intractable computations has revolutionized applied math-ematics. a Figure 1. D. More specifically, you want to test to. This same-side bias was first predicted in a physics model by scientist Persi Diaconis. 3. (uniformly at random) and a fair coin flip is made resulting in. Introduction The most common method of mixing cards is the ordinary riffle shuffle, in which a deck of ncards (often n= 52) is cut into two parts and the. ”It relates some series of card manipulations and tricks with deep mathematics, of different kinds, but with a minimal degree of technicity, and beautifully shows how the two. 8 percent chance of the coin showing up on the same side it was tossed from. View Profile, Richard Montgomery. This project aims to compare Diaconis's and the fair coin flip hypothesis experimentally. Figure 1 a-d shows a coin-tossing machine. , Holmes, S. Sunseri Professor of Mathematics and Statistics, Stanford University Introduction: Barry C. Diaconis and his colleagues carried out simple experiments which involved flipping a coin with a ribbon attached. This is one imaginary coin flip. The model asserts that when people flip an ordinary. With careful adjust- ment, the coin started. j satisfies (2. The bias, it appeared, was not in the coins but in the human tossers. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact. Born: 31-Jan-1945 Birthplace: New York City. It backs up a previous study published in 2007 by Stanford mathematician Persi Diaconis. Even if the average proportion of tails to heads of the 100,000 were 0. If a coin is flipped with its heads side facing up, it will land the same way 51 out of 100 times, a Stanford researcher has claimed. Measurements of this parameter based on high-speed photography are reported. Three academics—Persi Diaconis, Susan Holmes, and Richard Montgomery—through vigorous analysis made an interesting discovery at Stanford University. Diaconis–Holmes–Montgomery are not explicit about the exact protocol for flipping a coin, but based on [1, § 5. 1) Bet on whatever is face-up on the coin at the start of the flip. Holmes, G Reinert. To submit students of this mathematician, please use the new data form, noting this mathematician's MGP ID. Dynamical Bias in the Coin T oss! Persi Diaconis Susan Holmes à Richar d Montg omer y¤ Abstract. Suppose you want to test this. D. Gupta, Purdue University The production ofthe [MS Lecture Notes-MonographSeries isFlip a Coin Online: Instant coin to flip website | Get random heads or tails. Running away from an unhappy childhood led Persi Diaconis to magic, which eventually led to a career as a mathematician. In each case, analysis shows that, while things can be made approximately. They put it down to the fact that when you flip a coin off your thumb it wobbles, which causes the same side. Persi Diaconis, a former protertional magician who rubsequently became a profestor of statiatics and mathematics at Stanford University, found that a toesed coin that in caught in milais hat about a 51% chance of lasding with the same face up that it. A sharp mathematical analysis for a natural model of riffle shuffling was carried out by Bayer and Diaconis (1992). Holmes co-authored the study with Persi Diaconis, her husband who is a magician-turned-Stanford-mathematician, and. in mathematics from the College of the City of New York in 1971, and an M. Diaconis’ model proposed that there was a “wobble” and a slight off-axis tilt that occurs when. After you’ve got this down, we’ll look at a few ways to influence the outcome of the coin flip. The coin is placed on a spring, the spring is released by a ratchet, and the coin flips up doing a natural spin and lands in the cup. (b) Variationsofthe functionτ asafunctionoftimet forψ =π/3. 8. If you have additional information or corrections regarding this mathematician, please use the update form. Researchers Flipped A Coin 350,757 Times And Discovered There Is A “Right” Way To Call A Coin Flip. COIN TOSSING By PERSI DIACONIS AND CHARLES STEIN Stanford University Let A be a subset of the integers and let S. If they defer, the winning team is delaying their decision essentially until the second half. Lifelong debunker takes on arbiter of neutral choices: Magician-turned-mathematician uncovers bias in a flip of the coin by Esther Landhuis for Stanford Report. Sunseri Professor of Statistics and Mathematics at Stanford University. Advertisement - story. He had Harvard University engineers build him a mechanical coin flipper. 272 PERSI DIACONIS AND DONALD YLVISAKER If ii,,,,, can be normalized to a probability measure T,,,, on 0, it will be termed a distribution conjugate to the exponential family {Po) of (2. According to Diaconis’s team, when people flip an ordinary coin, they introduce a small degree of “precession” or wobble, meaning a change in the direction of the axis of rotation throughout. It backs up a previous study published in 2007 by Stanford mathematician Persi Diaconis. In a preregistered study we collected350,757coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by Persi Diaconis. . (2007). Persi Diaconis, a Stanford mathematician and practiced magician, can restore a deck of cards to its original order with a series of perfect shuffles. Persi Diaconis's publication list contains around 200 items. The findings have implications for activities that depend on coin toss outcomes, such as gambling. Ethier. Persi Diaconis is a mathematician and statistician working in probability, combinatorics, and group theory, with a focus on applications to statistics and scientific computing. He claims that a natural bias occurs when coins are flipped, which. Stanford mathematician Persi Diaconis published a paper that claimed the. , US$94. The Solutions to Elmsley's Problem. Do you flip a coin 50 50? If a coin is flipped with its heads side facing up, it will land the same way 51 out of 100 times, a Stanford researcher has claimed. 5. S. “I’m not going to give you the chance,” he retorted. Scientists shattered the 50/50 coin toss myth by tossing 350,757. Sunseri Professor of Statistics and Mathematics at Stanford University. Biography Persi Diaconis' Web Site Flipboard Flipping a coin may not be the fairest way to settle disputes. 182 PERSI DIACONIS 2. Discuss your favorite close-up tricks and methods. (2004). EN English Deutsch Français Español Português Italiano Român Nederlands Latina Dansk Svenska Norsk Magyar Bahasa Indonesia Türkçe Suomi Latvian. Suppose you want to test this. prediction from a physics model of human coin tossing developed by Diaconis, Holmes, and Mont-gomery (D-H-M; 2007). (PhotocourtesyofSusanHolmes. No coin-tossing process on a given coin will be perfectly fair. October 18, 2011. “Despite the widespread popularity of coin flipping, few people pause to reflect on the notion that the outcome of a coin flip is anything but random: a coin flip obeys the laws of Newtonian physics in a relatively transparent manner,” the researchers wrote in their report. It makes for facinating reading ;). , & Montgomery, R. The famous probabilist, Persi Diaconis, claims to be able to flip a fair coin and make it land heads with probability 0. 508, which rounds up perfectly to Diaconis’ “about 51 percent” prediction from 16 years ago. What is the chance it comes up H? Well, to you, it is 1/2, if you used something like that evidence above. View Profile, Susan Holmes. Diaconis' model proposed that there was a "wobble" and a slight off-axis tilt that occurs when humans flip coins with their thumb, Bartos said. 294-313. Throughout the. Persi Diaconis, a former professional magician who subsequently became a professor of statistics and mathematics at Stanford University, found that a tossed coin that is caught in midair has about a 51% chance of landing with the same face up that it. Regardless of the coin type, the same-side outcome could be predicted at 0. The chapter has a nice discussion on the physics of coin flipping, and how this could become the archetypical example for a random process despite not actually being ‘objectively random’. Point the thumb side up. Articles Cited by Public access. Room. Mathematicians Persi Diaconis--also a card magician--and Ron Graham--also a juggler--unveil the connections between magic and math in this well-illustrated volume. The same initial coin-flipping conditions produce the same coin flip result. PARIS (AFP) – Want to get a slight edge during a coin toss? Check out which side is facing upwards before the coin is flipped – then call that same side. Abstract We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck. The province of the parameter (no, x,) which allows such a normalization is the subject matter of the first theorem. With careful adjust- ment, the coin started heads up always lands heads up—one hundred percent of the time. However, it is not possible to bias a coin flip—that is, one cannot. However, naturally tossed coins obey the laws of mechanics (we neglect air resistance) and their flight is determined. Persi Diaconis, a former professional magician who subsequently became a professor of statistics and mathematics at Stanford University, found that a tossed coin that is caught in midair has about a 51% chance of landi ng with the same face up that it started wit h. , same-side bias, which makes a coin flip not quite 50/50. org. A specialty is rates of convergence of Markov chains. 00, ISBN 978-0-387-25115-8 This book takes an in-depth look at one of the places where probability and group theory meet. First, the theorem he refers to concerns sufficient statistics of a fixed size; it doesn’t apply if the summary size varies with the data size. With an exceptional talent and skillset, Persi. He breaks the coin flip into a. A prediction is written on the back (to own up, it’s 49). 20. Measurements of this parameter based on. The Diaconis–Holmes–Montgomery Coin Tossing Theorem Suppose a coin toss is represented by: ω, the initial angular velocity; t, the flight time; and ψ, the initial angle between the angular momentum vector and the normal to the coin surface, with this surface initially ‘heads up’. Regardless of the coin type, the same-side outcome could be predicted at 0. I am a mathematician and statistician working in probability, combinatorics, and group theory with a focus on applications to statistics and scientific computing. Suppose you want to test this. Persi Diaconis A Bibliography Compiled by. Categories Close-up Tricks Card Tricks Money & Coin Tricks Levitation Effects Mentalism Haunted Magic. 51 — in other words, the coin should land on the same side as it started 51 percent of the time. The coin toss in football is a moment at the start of the game to help determine possession. They have demonstrated that a mechanical coin flipper which imparts the same initial conditions for every toss has a highly predictable outcome – the phase space is fairly regular. Persi Diaconis had Harvard engineers build him a coin-flipping machine for a series of studies. Coin flipping as a game was known to the Romans as navia aut caput ("ship or head"), as some coins had a ship on one side and the head of the emperor on the other. The ratio has always been 50:50. Persi Diaconis 1. Stanford math professor and men with way too much time on their hands Persi Diaconis and Richard Montgomery have done the math and determined that rather than being a 50/50 proposition, " vigorously flipped coins tend to come up the same way they started. 3 Pr ob ability of he ads as a function of ψ . The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal. Here is a treatise on the topic from Numberphile, featuring professor Persi Diaconis from. (For example, changing the side facing up slightly alters the chances associated with the resulting face on the toss, as experiments run by Persi Diaconis have shown. His theory suggested that the physics of coin flipping, with the wobbling motion of the coin, makes it. ) Could the coin be close to fair? Possibly; it may even be possible to get very close to fair. Stanford University professor of mathematics and statistics Persi Diaconis theorized that the side facing up before flipping the coin would have a greater chance of being faced up once it lands. mathematician Persi Diaconis — who is also a former magician. The authors of the new paper conducted 350,757 flips, using different coins from 46 global currencies to eliminate a heads-tail bias between coin designs. Upon receiving a Ph. 5) gyr JR,,n i <-ni Next we compute, writing o2 = 2(1-Prof Diaconis noted that the randomness is attributed to the fact that when humans flip coins, there are a number of different motions the coin is likely to make. Persi Diaconis graduated from New York’s City College in 1971 and earned a Ph. Title. [0] Students may. If limn,, P(Sn E A) exists for some p then the limit exists for all p and does not depend on p. And they took high-speed videos of flipped coins to show this wobble. Ethier. tested Diaconis' model with 350,757 coin flips, confirming a 51% probability of same-side landing. At each round a pair of players is chosen (uniformly at random) and a fair coin flip is made resulting in the transfer of one unit between these two players. To test this, you spin a penny 12 times and it lands heads side up 5 times. Persi Diaconis, a former professional magician who subsequently became a professor of statistics and mathematics at Stanford University, found that a tossed coin that is caught in midair has about a 51% chance of landing with the same face up that it started with. As they note in their published results, "Dynamical Bias in the Coin Toss," laws of mechanics govern coin flips, meaning, "their flight is determined by their initial. Consider gambler's ruin with three players, 1, 2, and 3, having initial capitals A, B, and C units. [1] In England, this game was referred to as cross and pile. List of computer science publications by Persi Diaconis. Persi Diaconis UCI Chancellor's Distinguished Fellow Department of Mathematics Stanford University Thursday, February 7, 2002 5 pm SSPA 2112. Details. a 50% credence about something like advanced AI. For such a toss, the angular momentum vector M lies along the normal to the coin, and there is no precession. 2, No. Figures5(a)and5(b)showtheeffectofchangingψ. According to statistician Persi Diaconis, the probability of a penny landing heads when it is spun on its edge is only about 0. he had the physics department build a robot arm that could flip coins with precisely the same force. The coin toss is not about probability at all, its about physics, the coin, and how the “tosser” is actually throwing it. The bias was confirmed by a large experiment involving 350,757 coin flips, which found a greater probability for the event. We should note that the papers we list are not really representative of Diaconis's work since. Forget 50/50, Coin Tosses Have a Biasdarkmatterphotography - Getty Images. I cannot. "The standard model of coin flipping was extended by Persi Diaconis, who proposed that when people flip an ordinary coin, they introduce a small degree of 'precession' or wobble – a change in. This slight. Persi Diaconis. 8 per cent likely to land on the same side it started on, reports Phys. According to math professor Persi Diaconis, the probability of flipping a coin and guessing which side lands up correctly is not really 50-50. Indeed chance is sometimes confused with frequency and this. And because of that, it has a higher chance of landing on the same side as it started—i. 1. Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. , Montgomery, R. It is a familiar problem: Any. The outcome of coin flipping has been studied by the mathematician and former magician Persi Diaconis and his collaborators. ” See Jaynes’s book, or any of multiple articles by Persi Diaconis. We show that vigorously flipped coins tend to come up the same way they started. he had the physics department build a robot arm that could flip coins with precisely the same force. But to Persi, who has a coin flipping machine, the probability is 1. Researchers from the University of California, Berkeley, conducted a preregistered study to test the prediction of a physics model of human coin tossing developed by Persi Diaconis. In a preregistered study we collected 350,757 coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by Diaconis, Holmes, and Montgomery (D-H-M; 2007). According to math professor Persi Diaconis, the probability of flipping a coin and guessing which side lands up correctly is not really 50-50. What happens if those assumptions are relaxed?. In 1962, the then 17-year-old sought to stymie a Caribbean casino that was allegedly using shaved dice to boost house odds in games of chance. Math Horizons 14:22. We analyze the natural process of flipping a coin which is caught in the hand. We analyze the natural process of flipping a coin which is caught in the hand. Introduction Coin-tossing is a basic example of a random phenomenon. In 2007,. Now that the issue of dice seems to have died down a bit anyone even remotely interested in coin flipping should try a google search on Persi Diaconis. Explore Book Buy On Amazon. Diaconis, P. In 2007 the trio analysed the physics of a flipping coin and noticed something intriguing. 49, No. Everyone knows the flip of a coin is a 50-50 proposition. Sunseri Professor of Statistics and Mathematics at Stanford University and is particularly known for tackling mathematical problems involving randomness and randomization, such as coin flipping and shuffling playing cards. $egingroup$ @Michael Lugo: Actually, according to work of Persi Diaconis and others, it's hard to remove the bias from the initial orientation of the coin. Publications . COIN TOSSING BY PERSI DIACONIS AND CHARLES STEIN Stanford University Let A be a subset of the integers and let Snbe the number of heads in n tosses of a p coin. Persi Diaconis, a former professional magician who subsequently became a professor of statistics and mathematics at Stanford University, found that a tossed coin that is caught in midair has about a 51% chance of landing with the same face up that it. Persi Diaconis did not begin his life as a mathematician. 5] here is my version: Make a fist with your thumb tucked slightly inside. More specifically, you want to test to determine if the probability that a coin that starts out heads up will also land heads up is more than 0. Persi Diaconis left High School at an early age to earn a living as a magician and gambler, only later to become interested in mathematics and earn a Ph. He’s going to flip a coin — a standard U. perceiving order in random events. List price: $29. Further, in actual flipping, people exhibit slight bias – "coin tossing is. To figure out the fairness of a coin toss, Persi Diaconis, Susan Holmes, and Richard Montgomery conducted research study, the results of which will entirely change your view. W e analyze the natural pro cess of ßipping a coin whic h is caugh t in the hand. 8% of the time, confirming the mathematicians’ prediction. Another Conversation with Persi Diaconis David Aldous Abstract. Persi Diaconis Mary V. The team took a herculean effort and got 48 people to flip 350,757 coins from 46 different countries to come up with their results. the conclusion. However, a study conducted by American mathematician Persi Diaconis revealed that coin tosses were not a 50-50 probability sometime back. We analyze the natural process of flipping a coin which is caught in the hand. We show that vigorously flipped coins tend to come up the same way they started. In each case, analysis shows that, while things can be made approximately. the conclusion. Persi Diaconis. 828: 2004: Asymptotics of graphical projection pursuit. Room. A specialty is rates of convergence of Markov chains. One way to look for the line would be to flip a coin for the duration of our universe’s existence and see what the longest string of Heads is. The Mathematics of the Flip and Horseshoe Shuffles. They have demonstrated that a mechanical coin flipper which imparts the same initial conditions for every toss has a highly predictable outcome – the phase space is fairly regular. If a coin is flipped with its heads side facing up, it will land the same way 51 out of 100 times, a Stanford researcher has claimed. Dynamical bias in the coin toss SIAM REVIEW Diaconis, P. 1 shows this gives an irreducible, aperi- odic Markov chain with H,. We develop a clear connection between deFinetti’s theorem for exchangeable arrays (work of Aldous–Hoover–Kallenberg) and the emerging area of graph limits (work of Lova´sz and many coauthors). Our analysis permits a sharp quantification of this: THEOREM2.