Ars Conjectandi

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The cover page of Ars Conjectandi
The cover page of Ars Conjectandi

Ars Conjectandi (Latin: The Art of Conjecturing) is a mathematical paper written by Jakob Bernoulli, published posthumously in 1713. The work consolidated existing probability theory at the time, as well as adding new material to the subject. It has been dubbed a landmark in the subject by author William Dunham.

Bernoulli wrote the text between 1684 and 1689 to encompass the existing work on probability, which included the work of mathematicians such as Christian Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. It also incorporated topics such as his theory of permutations and combinations, as well as those more distantly connected to number theory: the derivation and properties of the Bernoulli numbers.

Contents

[edit] Background

Gerolamo Cardano, a pioneer of probability theory
Gerolamo Cardano, a pioneer of probability theory

The subject of probability in Europe was first formally developed in the sixteenth century with the work of Cardano, whose interest in probability was largely due to his habit of gambling.[1] He formalized what is now known as the classical definition of probability: if an event has a possible outcomes and we select any b of those such that b ≤ a, the probability of any of the b occurring is \begin{smallmatrix}\frac{b}{a}\end{smallmatrix}. He was the first mathematician to compute a theoretical probability (as opposed to an empirical one), but his actual influence was not great. Though he wrote a book on the subject in 1525 entitled Liber de ludo aleae (Book on Games of Chance), it was not published until after his death in 1663.[2][3]

The date which historians cite the "beginning" of probability in its modern sense is 1654, when Pascal and Fermat began a correspondence discussing probability. This was initiated because in that year, a gambler from Paris named Antoine Gombaud sent Pascal, as well as other mathematicians, several questions on probability. Pascal and Fermat's correspondence interested other mathematicians as well, including Christian Huygens, who in 1657 published De ratiociniis in aleae ludo (Calculations in Games of Chance).[2] During this period, Pascal also published his results on the triangle that bears his name today; Pascal's triangle, which referred to in his work Traité du triangle arithmétique (Traits of the Arithmetic Triangle) as the "arithmetic triangle".[4] Later, Jan de Witt published similar material in his 1671 work Waerdye van Lyf-Renten (A Treatise on Life Annuities), which used statistical concepts to determine life expectancy.[5]

Jakob Bernoulli wrote the work encompassing these between the fertile years 1684 and 1689; his output in terms of mathematical research was great during that time.[1] When he began the work in 1684 at the age of 30, he had not yet read Pascal's work on the "arithmetic triangle" nor de Witt's work on statistical probability. He had earlier requested a copy of the latter from his acquaintance Gottfried Leibniz, but Leibniz failed to provide it. Leibniz, however, did provide Pascal's and Huygen's work, on which Bernoulli based his Ars Conjectandi.[6] Bernoulli's nephew Nicholas published the manuscript in 1713 after Bernoulli's death in 1705.[7][8]

[edit] Contents

Abraham de Moivre, who was influenced by Bernoulli's work
Abraham de Moivre, who was influenced by Bernoulli's work

Bernoulli's work, which is divided into four parts,[6] covered most notably his theory of permutations and combinations; the standard foundations of combinatorics today. It also covered Bernoulli numbers, which were related more to number theory than probability. These bear his name today, and are one of his more notable achievements.[9][10]

In the first part, Bernoulli discussed Huygen's De ratiociniis in aleae ludo in depth and solved the problems Huygens had posed at the end.[6] Bernoulli particularly developed Huygen's concept of expected value. Huygens had developed the formula E=\frac{p_0a_0+p_1a_1+p_2a_2+\ldots+p_na_n}{p_0+p_1+\ldots+p_n},[11] E is the expected value, pi are the probabilities of attaining each value, and ai are the attainable values. Bernoulli normalized the expected value by assuming that pi are the probabilities of all the disjoint outcomes of the value, thus leading to the fact that p0+p1+...+pn=1. Another key theory developed in this part was the probability achieving at least a number of successes from a number of events with multiple outcomes given that the probability of success in each was the same. Bernoulii showed through mathematical induction that given that a was the number of favorable outcomes in each event, b was the number of total outcomes in each event, d was the desired number of successful outcomes, and e was the number of events, the probability was P=\sum_{i=0}^{e-d}\binom{e}{d+i}\left(\frac{a}{b}\right)^{a+v}\left(\frac{b-a}{b}\right)^{e-d-i}.[12] This part also discussed what is now known as the Bernoulli distribution.[13]

The second part discussed combinatorics, or the systematic numeration of objects—it was in this part that the permutations and combinations that would form the basis of the subject were introduced. It also discussed the Bernoulli numbers. These rational numbers were the coefficients of the expansion of \begin{smallmatrix}\frac{x}{1-e^{-x}}\end{smallmatrix} as an exponential series.[14] Additionally, this part also contained Bernoulli's formula for the sum of powers of integers, which influenced Abraham de Moivre's work later.[13]

In the third part, Bernoulli applied the discussed probability techniques to the common chance games of the day—games played with cards or dice.[6] He presented probability problems related to these. In addition, he poser generalizations of the problems without specific constants. For example, a problem involving the expected number of "court cards" one would pick from a deck of 20 cards containing 10 court cards could be generalized to a deck with a cards that contained b court cards such that b<a.[15]

The fourth part discusses applying probability to civilibus, moralibus, and oeconomicis, or to personal, judicial, and financial decisions. In this section, Bernoulli uses the term "probability" in the modern sense for the first time in the work—before this, the use of combinatorial methods to determine theoretical chances of outcomes of an event were referred to as "equity," and "probability" was used in a strictly empirical sense.[6] This section contained most notably a result resembling the law of large numbers, which Bernoulli described as predicting that empirical probability would approach theoretical probability as more trials were held.[7] This early version of the law is known today as either Bernoulli's theorem or the weak law of large numbers.[16]

Bernoulli also appended to Ars Conjectandi a tract on calculus. It concerned in particular infinite series.[13]

[edit] Legacy

Colin Maclaurin
Colin Maclaurin

Dunham called Ars Conjectandi as "the next milestone of probability theory [after the work of Cardano]," as well as "Jakob Bernoulli's masterpiece."[1] It greatly aided what Dunham describes as "Bernoulli's long-established reputation [...]".[17]

Bernoulli's work influenced many contemporary and subsequent mathematicians. The tract on calculus has been quoted frequently; most notably by the Scottish Colin Maclaurin. Abraham de Moivre was particularly influenced by Bernoulli's work. He wrote on the concept of probability in The Doctrine of Chances.[18] De Moivre's most notable achievement in probability was the central limit theorem, by which he was able to approximate the binomial distribution. He did this using an asymptotic sequence for the factorial function—which he had developed with James Stirling—and Bernoulli's formula for the sum of powers of numbers.[13]

Thomas Simpson achieved a similar result that closely resembled de Moivre's. Thomas Bayes wrote an essay discussing theological implications of de Moivre's results.[13]

[edit] See also

[edit] Notes

  1. ^ a b c (Dunham 1990, p. 191)
  2. ^ a b Abrams, William, A Brief History of Probability, Second Moment, <http://www.secondmoment.org/articles/probability.php>. Retrieved on 23 May 2008 
  3. ^ O'Connor, John J. & Robertson, Edmund F., Cardano Biography, MacTutor, <http://www-history.mcs.st-andrews.ac.uk/Biographies/Cardan.html>. Retrieved on 23 May 2008 
  4. ^ “Blaise Pascal”, Encyclopædia Britannica Online, 2008, <http://www.britannica.com/EBchecked/topic/445406/Blaise-Pascal/15001/Pascals-life-to-the-Port-Royal-years#ref=ref365130>. Retrieved on 23 May 2008 
  5. ^ (Brakel 1976, p. 123)
  6. ^ a b c d e (Shafer 2006, pp. 3–4)
  7. ^ a b (Bernoulli 2005, p. i)
  8. ^ Weisstein, Eric, Bernoulli, Jakob, Wolfram, <http://scienceworld.wolfram.com/biography/BernoulliJakob.html>. Retrieved on 9 June 2008 
  9. ^ “Jakob Bernoulli”, Encyclopædia Britannica Online, 2008, <http://www.britannica.com/EBchecked/topic/62599/Jakob-Bernoulli#ref=ref782754>. Retrieved on 23 May 2008 
  10. ^ “Bernoulli”, The Columbia Electronic Encyclopedia (6th ed.), 2007 
  11. ^ The notation \begin{smallmatrix}\binom{n}{r}\end{smallmatrix} represents the number of ways to choose r objects from a set of n distinguishable objects without replacement.
  12. ^ (Schneider 2006, pp. 7–8)
  13. ^ a b c d e (Schneider 2006, p. 1)
  14. ^ (Maseres, Bernoulli & Wallis 1798, p. 115)
  15. ^ (Hald 2003, p. 254)
  16. ^ Eric W. Weisstein, Weak Law of Large Numbers at MathWorld.
  17. ^ (Dunham 1990, p. 192)
  18. ^ (de Moivre 1716, p. i)

[edit] References

  • Bernoulli, Jakob (2005), On the Law of Large Numbers, Part Four of Ars Conjectandi (English translation), ISBN 3-938417-14-5 
  • Brakel, J. van (1976), “Some Remarks on the Prehistory of the Concept of Statistical Probability”, Archive for History of Exact Sciences (Heidelberg) 16 (2), ISSN 0003-9519 
  • Dunham, William (1990), Journey Through Genius (1st ed.), John Wiley and Sons, ISBN 0-471-50030-5 
  • Hald, Anders (2005), A History of Probability and Statistics and Their Applications Before 1750, Wiley, ISBN 978-0-471-47129-5 
  • Maseres, Francis; Bernoulli, Jakob & Wallis, John (1798), The Doctrine of Permutations and Combinations, British Critic 
  • de Moivre, Abraham (1716), ISBN 978-0821821039 
  • Schneider, Ivo (2006), “Direct and Indirect Influences of Jakob Bernoulli's Ars Conjectandi in 18th Century Great Britain”, Electronic Journal for the history of Probability and Statistics 2 (1) 
  • Shafer, Glenn (2006), The Significance of Jacob Bernoulli’s Ars Conjectandi for the Philosophy of Probability Today, Rutgers University 

[edit] External links