The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern

by Keith Devlin

Before the mid-seventeenth century, scholars generally agreed that it was impossible to predict something by calculating mathematical outcomes. One simply could not put a numerical value on the likelihood that a particular event would occur. Even the outcome of something as simple as a dice roll or the likelihood of showers instead of sunshine was thought to lie in the realm of pure, unknowable chance.The issue remained intractable until Blaise Pascal wrote to Pierre de Fermat in 1654, outlining a solution to the “unfinished game” problem: how do you divide the pot when players are forced to end a game of dice before someone has won? The idea turned out to be far more seminal than Pascal realized. From it, the two men developed the method known today as probability theory.

In The Unfinished Game, mathematician and NPR commentator Keith Devlin tells the story of this correspondence and its remarkable impact on the modern world: from insurance rates, to housing and job markets, to the safety of cars and planes, calculating probabilities allowed people, for the first time, to think rationally about how future events might unfold.

Genres: History, Mathematics, Science, Nonfiction, Education, Biography, 17th Century

208 pages, Hardcover

First published January 1, 2008

About the author

Dr. Keith Devlin is a co-founder and Executive Director of the university's H-STAR institute, a Consulting Professor in the Department of Mathematics, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 26 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is "the Math Guy" on National Public Radio.

Reviews

[Mircea Petcu · Rating 5 out of 5]

Ma gandesc la casele de pariuri si salile de jocuri care au impanzit Romania in ultimii ani:

"O alta mostenire a acelei corespondente este industria cazinourilor. Este cu adevarat o industrie, si inca una foarte profitabila, deoarece cazinourile nu joaca jocuri de noroc- lasa asta in seama clientilor. Folosind capacitatea teorie probabilitatilor de a prezice rezultatele jocurilor de noroc, ele pot calcula dinainte si cu precizie cati bani vor castiga in fiecare saptamana. Sondajele politice se insala uneori, dar cazinourile nu se insala niciodata."

[S. · Rating 4 out of 5]

The book illustrates the history of probability, not dense with equation's, in fact all of which are made easier to understand with all the examples and explanations that the author used.
Why don't we go on learning maths this way, fun, simple and tightly related to the 'real' world ?

[Razvan Zamfirescu · Rating 4 out of 5]

Spicuiri din recenzia finala care se gaseste pe blogul meu



..........................................

După Împotriva Zeilor, Partida neterminată a lui Devlin a apărut foarte oportun pentru mine. Cu ceva semne de întrebare și dorință de a afla mai multe despre relația dintre Fermat și Pascal, nu am ezitat absolut deloc când am văzut volumul la raftul de reeditări.

Nu mi-a părut rău absolut deloc. Foarte multe informații le aveam deja din cartea lui Bernstein, dar Devlin se ocupă mult mai detaliat de discuția dintre Fermat și Pascal cu privire la problema împărțirii potului între jucătorii care se opresc din joc în momentul în care unul dintre ei conduce cu doi la unu.

Devlin își dezvoltă cartea luând pasaje din scrisoare lui Pascal către Fermat și așa se și formează capitolele. Fiecare capitol are parte de o scurtă trecere în revistă a următorului pas presupus de fragmentul inaugural, observații cu privire la evoluția probabilității în istoria umanității, scurte informații despre biografia celor doi mari matematicieni precum și formule matematice menite a explica și face cât se poate de accesibilă discuția și teoria gânditorilor francezi.

O cărțulie cu un ton didactic, care pe mine m-a bucurat pentru că nu sunt un foarte priceput într-ale matematicii, ton care nu cred că-i va atrage pe foarte mulți cunoscătorii sau pasionați de acest domeniu pentru că, până la urmă, Devlin nu face altceva decât să adune la un loc și să expună cât mai simplist una dintre cele mai interesante discuții care au existat în istoria matematicii. Cu alte cuvinte este un fel de Idiotʼs Guide to Pascal letters to Fermat.

O recomand celor ca mine.

.......................................

[Valerie · Rating 5 out of 5]

Letter writing is most certainly a lost art. "I beg you to inform me how you would proceed in your research on this problem. I shall receive your reply with respect and joy, even if your opinion should be contrary to mine." contrasted with today's modern flame wars. Having just taken a class in decision quality at Stanford, I found the discussions of Baye's formula and assessing risk by using probability very interesting. It is difficult to imagine a time before probability mathematics.

[Beauty Mary · Rating 4 out of 5]

Having studied the theory of probabilities at University, it was really nice learning some new information and ideas about this subject. I can remember how passionate our Maths teacher was about Pascal and Fermat's letters and he was the one who introduced us to this book. Of course, like every other student, I didn't read it at that time, my mind being focused elsewhere, but I am glad that I've finally managed to read this book.

In fact, I loved every class that was somehow related to the theory of probabilities: Game Theory, Decision Theory etc. and with time, I got very involved in my studies on these particular subjects. Very interesting indeed how our views and life choices can change when you put numbers to them. Absolutely fascinating.

[Marius Bancila · Rating 3 out of 5]

The title is a little bit misleading as the book is not entirely about the letters Pascal and Fermat have exchanged in 1654, but rather a history of the science of probabilities that started with the problem of the unfinished game. Devlin does focus on the letters of the two great French mathematicians but also shows how others have drawn inspiration from the methods Pascal and Fermat have established and how they developed and applied math to real world problems (not only gaming). The book is nicely spiced with short biographies of many mathematicians. Overall Devlin makes a good case showing how hard it was for the great thinkers of 17th century to overstep what seemed rationale and develop correct mathematics for determining the chances of an event to occur.

[Dan Dumitrescu · Rating 4 out of 5]

Savuros sa vezi cum doi pasionati de matematica (fara a fi matematicieni), plecand de la studiul jocurilor de noroc pun bazele la ceea ce urma sa devina teoria probabilitatilor, urmand a fi folosita nu numai in afacerile cu jocuri de noroc dar si in bursa, asigurari si de fapt, in toate domenile care incercau sa-si previzioneze evolutia.

De citit, de preferat inainte, Marea teoremă a lui Fermat, pentru a intelege respectul lui Pascal pentru Fermat.

Pascal reuseste chiar sa vina cu o formula care se afirme rationalitatea credintei in Dumnezeu "p x ∞ + (1 - p) x Z", concluzia sa principala este ca “suntem obligati sa pariem”ca Dumnezeu exista.

[Elizabeth · Rating 3 out of 5]

Not as entertaining or efficiently informative as The Man of Numbers. Felt a bit disjointed and speculative at times, as though Devlin thought, "Oh, yes, one other thing while I'm at it" many times throughout.

[William Schram · Rating 4 out of 5]

The Problem of Points taunted humanity for centuries. A letter between Blaise Pascal and Pierre de Fermat changed all of that forever.

In "The Unfinished Game," Keith Devlin goes over the paradigm-shattering piece of correspondence that introduced the world to Probability Theory. Devlin explains the reasoning of each side and the points that each person made in previous letters. The game-changing idea was to predict what would happen if the game continued. People assumed that games of chance were all up to the gods and did not realize that it was possible to predict events.

Devlin discusses other aspects of Probability Theory as well. For example, in the same period as the correspondence, John Graunt developed a method to read statistics from raw data. Graunt was surprisingly accurate for the time and used for decades to make actuarial predictions.

We can't discuss this era without some mention of the Bernoulli family, and we get that too. Finally, Devlin talks a bit about Bayes' Rule. Conditional Probability was far too tedious before computers were widespread.

Devlin does not include too many equations in the book. He mainly discusses the methods employed.

[Vishal Katariya · Rating 3 out of 5]

Nice history of probability theory.

[Kelly Vincent · Rating 5 out of 5]

This book tells the story of the origins of probability, which emerged more recently than you would expect for such a fundamental field. It's somewhat famous among certain types of nerds that probability theory came from a handful of mathematicians pondering certain types of gambling. Specifically, the real origins are documented in a series of letters between Blaise Pascal and Pierre de Fermat. So this book includes selections from the (extant) letters and explains and discusses them, giving the historical context and then continuing on to what the letters inspired. It's an interesting book, and it's always cool to actually see the primary sources that history books are relying on. It's also a fairly slim book. The math isn't extremely difficult to follow (this is no "calculus-based probability and statistics" textbook) and I don't think a full understanding of all of it is critical to an appreciation of the book, anyway.

Prior to the events kicked off by the exchange of letters, virtually everyone thought the future was impossible to predict (well, in a secular sense, anyway). So while some avid gamblers tried to pay attention to the behavior of the games they played to figure out how to best bet, it seemed rather pointless. The late-fifteenth century mathematician Pacioli published a book that included the "problem of the unfinished game" (also called the "problem of the points"), which asks what the fair way to distribute the money when a game has to be aborted before its intended conclusion. For instance, if two players decide to play the best of 7 but have to stop after only 4 games, with 1 player with 1 win and the other with 3, how much of the pot should each walk away with? The only really obvious scenario is when they are exactly tied. All others seemed unknowable.

The Chevalier de Mere was good enough at calculating odds that he used the modern casino long haul strategy of identifying games with probability just slightly in his favor (i.e., around 51% for him) and just playing a whole bunch of times. He contacted Pascal with several questions relating to games of chance, including the problem of the unfinished game. Pascal tried to solve it and then wrote to Fermat to get a second opinion on his solution. Thus, the exchange.

Devlin walks us through the letters, Pascal's solution, Fermat's superior solution, and Fermat's patient attempts to explain the revolutionary ideas embedded in his solution to Pascal, who struggled to grasp it. I found this pretty interesting, because Pascal is one of the Greats, yet he just could not wrap his head around what Fermat was proposing. Sometimes math is hard, even when you're good at it. This certainly fits with my own experience of trying to learn probability theory--it is probably (heh) as counter-intuitive as any accurate idea can be. The most novel part of Fermat's approach, the one Pascal had so much trouble accepting because it seemed illogical, is the idea that when you are calculating probabilities on a game of say, best of 7, you have to calculate the odds based on 7 plays--not just 5 if one player would have won and the game would have finished. So with the unfinished game problem, if one player is leading after 3 plays, you still have to calculate the odds on 7 plays to get the right answers.

Devlin goes on to discuss what happened with the new ideas. This includes (among other things): the first real instance of applied statistics (a 1662 pamphlet analyzing mortality rates in London), the landmark text that Huygens wrote on probability theory (starting with Fermat and Pascal but adding his own), the (amazing and numerous) Bernoullis, psychology and risk, probability in courts, the Normal distribution, annuities and insurance, Bayes, Site Profiler (the DoD system that predicted the events of 9/11, along with many other incidents that didn't happen), DNA profiling, and financial derivatives.

[Claudia · Rating 4 out of 5]

It will never cease to amaze me how few hundred years ago, when relying almost exclusively on their minds, people where able to develop and discover such things. This book details how the theory of probabilities emerged from the question on how the pot should be divided between two players, when one of them leads with 2 to 1 and they stop playing.

Keith Devlin, professor of mathematics at Stanford University, dissects the correspondence between Blaise Pascal and Pierre de Fermat and explains, in a very simple and understandable way, all the thinking and connections which led to that discovery and, later on, to statistics, risk management and many other predictions used today, from financial trends to terrorist’s attacks. He also reviews the roles others had in this matter, which are not as famous as these two, but had, nonetheless, a major contribution to this particular field.

And the letter between the two is just a piece of art… too bad these days we resume ourselves to electronic succinct messages.

A highly enjoyable and interesting book.

[Kelly Jackson · Rating 4 out of 5]

I do have a math background for disclosure purposes. I found the subject matter and historical context interesting. The writing style was a little grating. There was a lot of condescension in the prose. It's OK if you don't understand it...

[John Landis · Rating 1 out of 5]

A complete waste of my time. I would rather they chose either the math or the people involved to concentrate on instead of focusing on both and failing twice.

[Fred Cheyunski · Rating 4 out of 5]

Basis of Probability/Statistics for Our Modern World (and Limitations) - In under 200 pages, Keith Devlin goes over the key letter of Pascal to Fermat in 1654, which is well known by mathematicians but not usually read; the letter is included, related history and implications are discussed in very readable manner.

That is, in a Preface and 10 chapters, the book goes into the origins of probability, statistics, and the modern world; it addresses the ability to predict the future, or at least the likelihood of different events based on similar ones in the past. Pascal’s correspondence with Fermat began with the problem of how to award points if a game of chance could not be completed as first stated earlier in the 15th century. The book also provides fascinating background on Pascal and Fermat as well as showing how the use of numbers progressed from Fibonacci (for additional information see my review of Devlin’s book on this subject that appeared in 2012).

Among my favorite parts of the book are those that talk about the development of our ability to quantify risk which made insurance, liquid capital markets, and global corporations like Amazon, Google, and Microsoft possible. There is the discussion of Graunt’s work on mortality rolls during rein of Charles II (1676) to help warn about the spread of bubonic plague and bring probability theory out of the gaming room. Then, Huygens’s formalization/improvement of methods, and advances by Jacob Bernoulli with his 1713 “law of large numbers” enabled sampling and prediction. Also, receiving attention are de Moivre discovery of the bell curve in1733 as well as Gauss’s and Bayes’ later contributions. Devlin even mentions the Black Scholes formula which permitted financial derivatives valuation (and the Nobel economics prize in 1997) that lead to so much difficulty in 2008 around the time this book first appeared (due to the publication date we can forgive the author for more attention to such ills).

Along the lines of probability and statistics underlying assumptions and limitations, a good companion to this book could be Nazism Taleb’s “The Black Swan: Second Edition: The Impact of the Highly Improbable” (2012). Some of my favorite parts from that book include recognizing narrative’s strength in promoting and storytelling for completion and being wary of its use for prediction. The added segments on the four quadrants, when to use typical statistics, and when to look for black swans or improbable events is also most pertinent.

So, for a good read on the basis of probability and statistics for the modern world consider “The Unfinished Game.”

[Jim D'Ambrosia · Rating 3 out of 5]

Well, of course that title of one letter making the world modern is quite the exaggeration. Decent book. Short. The theme is the development of probability theory in mathematics. The problem of the unfinished game is this: suppose you and a friend are playing some game, with some money at stake. You are ahead, but the two of you have to break off the game, for whatever reason. How should you divide the pot? It's a fairly basic question these days, but at the time, none of the theory to solve it had been developed, until Pascal and Fermi put their heads to it.

It seems like Devlin likes to stretch out his writing maybe a little too finely; this was also the case in his book, The Man of Numbers. But that didn't bother me too much. The book is very readable, and provides a good introduction to the early development of probability theory. There aren't enough books on the history of mathematics that really tie the development of a theory to the real-world situations the theory was meant to solve.

[Ramona Cantaragiu · Rating 3 out of 5]

Of course the title is an exaggeration as it is also shown in the book, but if it makes more people interested in the history if maths, i can let it slide. The first half of the book is mostly about what the title promises, i.e. the correspondence between Pascal and Fermat regarding the allocation of winnings if a game of chances was not finished by the parties. Reading their letters really made it evident that letter writting is a beautiful and refuned but lost art. The second part of the book introduces other mathematicians who have brought their contributions to probability theory and it made for a less interesting read since it was bit rushed. You need some math knowledge to understand what Devlin is talking about but you can skip parts and still get the main point that it was difficult for people in the seventeenth century to grapple with the idea that events in the future could be predicted when all was supposed to be God's will.

[Sarah · Rating 3 out of 5]

Three and a half. This is a wonderful general-readership introduction to the epistolic relationship between two brilliant French mathematicians, Pascal and Fermat, which led to the development of probability theory in the West. The analysis is insightful, the explanations simple and clear. The most critical of the letters from 1654 is included (in English translation) in the back of the book. I would have liked to have read it first in it entirety, but, that's fine. Further research and investigation has led me to the understanding that Pascal and Fermat, though revolutionary in Europe, were making an independent (re)discovery of probability (precursor) that was well-established in India and possibly the Middle East, prior. No mention of that here, a disappointing but familiar Euro-centric take.

Cited in Money: The True Story of a Made-Up Thing, ch. 4. Borrowed from NOBLE network.

[Xin · Rating 3 out of 5]

First Mathematics history book I have encountered. Definitely the most nerdy book I have read. The premise of the book is about how Pascal and Fermat discovered the earliest version of probability through the “game of points.” Can’t help but to notice Pascal is a genius, but tragically died young.

The content is inherently interesting, but became more repetitive as there is only one central theme. I did however have a good conversation with the neighbor math professor, who is excited about using the game of points to teach his students about expected value next week in his class.

The book overall is so-so, but I can tell the author is more excited about the content than the average reader. I can admire people who love what they do.

[Jeroen · Rating 4 out of 5]

An easy to read book on the origin of probability theory. The theory is explained well and established based on a letter between Pascal and Fermat. Easy to read for people that don't already know some of the probability theory basics and interesting background reading for those who do. Main drawback of the book is the easy way the author uses historic dates to set context of developments after the start by Fermat/Pascal. Those dates sometimes feel a bit chaotic and unstructured.

[Matthew · Rating 1 out of 5]

Enough content to maybe fill a Guardian Long Read.

The first bit of the book explains how discovering probability was one of the greatest mathematical achievements of all time; only to go on and say that all we’ve managed to do with that knowledge is create multi-billion dollar industries out of gambling and insurance.

[Ezra · Rating 2 out of 5]

It just seems to me that Devlin gives so much credit to this Pascal-Fermat correspondence and frames it as a philosophical revolution, but the evidence he presents paints a different picture of the development of basic probability and statistics in which the Pascal-Fermat correspondence is only one of many historical developments, and perhaps not even the most important one.

[Peter Ypma · Rating 3 out of 5]

De eerste zes hoofdstukken zijn heel goed en daarin wordt heel goed de brief en alles wat daarbij hoort beschreven. De reden dat ik op 3 sterren uitkom, is het tweede deel van het boek. Die gaan niet echt meer over de brief en die hoofdstukken voelen voor mij wat minder af (alsof de schrijver nog pagina's over had en dus ook nog maar wat over half gerelateerde problemen ging schrijven).

[Sonya Mann · Rating 4 out of 5]

Pretty good

Moderately entertaining and fairly informative. But not exceptionally written. I know a little more about the origins and anatomy of probability theory, so that makes me happy.

[Kiora · Rating 5 out of 5]

Super quick read with introductions to a lot of mathematicians that influenced probability theory. My favorite part was how he delved into the thinking they had to overcome to get to the breakthrough.

[Nikos Koukis · Rating 1 out of 5]

Meh, not really interesting.

The only fact worth knowing is that Fermat and Pascal developed the modern theory of probabilities in 1654 via a series of mail correspondences. There you go. I saved you from going through 200 boring pages

[Carter · Rating 3 out of 5]

Some of the world's most ground breaking contributions to probability theory, emerged historically, in correspondence, in a period, when it was more difficult to travel. I would love, to take some time to look at some of the primary source materials, at some point. Recommended.

[Elliot · Rating 3 out of 5]

This book is a recounting of the history of probability theory. It's reasonably well written and not too long. It's alright, but it gets brought down by some weird things like the god-like pedestal that the author place Fermat on.

[Brett · Rating 3 out of 5]

Pascal and Fermat are two fascinating characters in the history of math. However, I simply wasn't interested in the story here.

[John Hopkins]

I put it aside without finishing. I would recommend for any actual mathematician, though.