Definitions

# Ars Conjectandi

Ars Conjectandi (Latin: The Art of Conjecturing) is a mathematical paper written by Jakob Bernoulli and published eight years after his death by his nephew, Niklaus Bernoulli, in 1713. The work both consolidated existing probability theory and added to the subject. It has been dubbed a landmark in the subject by popular mathematical historian William Dunham. It also influenced contemporary and later mathematicians, such as Abraham de Moivre.

Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christian Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. It 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, for instance. Other core topics from probability, such as expected value were also included.

## Background

In Europe, the subject of probability 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. He formalized what is now called 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 $begin\left\{smallmatrix\right\}frac\left\{b\right\}\left\{a\right\}end\left\{smallmatrix\right\}$. but his actual influence was not great; he wrote only one book on the subject in 1525 entitled Liber de ludo aleae (Book on Games of Chance), though it was not published until after his death in 1663.

The date which historians cite as 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, and other mathematicians, several questions on probability; in particular he posed the problem of points, concerning a theoretical two-player game in which a prize must be divided between the players due to external circumstances halting the game. Pascal and Fermat's correspondence interested other mathematicians, including Christian Huygens, who in 1657 published De ratiociniis in aleae ludo (Calculations in Games of Chance). During this period, Pascal also published his results on the Pascal's triangle. He referred to the triangle in his work Traité du triangle arithmétique (Traits of the Arithmetic Triangle) as the "arithmetic triangle". 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.

Bernoulli produced a great deal of mathematical output between 1684 and 1689 including Ars Conjectandi. 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 Ars Conjectandi is based. Bernoulli titled the work Ars Conjectandi because he wished to link it to the concept of ars inveniendi from scholasticism, which in turn would indicate that his results could be applied to all facets of society and life. His nephew Nicholas published the manuscript in 1713 after Bernoulli's death in 1705.

## Contents

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

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. Bernoulli particularly developed Huygen's concept of expected value, or the weighted average of all possible outcomes of an event. Huygens had developed the following formula:

$E=frac\left\{p_0a_0+p_1a_1+p_2a_2+cdots+p_na_n\right\}\left\{p_0+p_1+cdots+p_n\right\}.$
In this formula, 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, today called Bernoulli trials, with multiple outcomes given that the probability of success in each was the same. Bernoulli 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 could be expressed as

$P=sum_\left\{i=0\right\}^\left\{e-d\right\}binom\left\{e\right\}\left\{d+i\right\}left\left(frac\left\{a\right\}\left\{b\right\}right\right)^\left\{a+v\right\}left\left(frac\left\{b-a\right\}\left\{b\right\}right\right)^\left\{e-d-i\right\}.$

The first part also discussed what is now known as the Bernoulli distribution.

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 general formula for sums of integer powers; the free coefficients of this are therefore called the Bernoulli numbers, which have proven widely useful in number theory. Additionally, this part also contained Bernoulli's formula for the sum of powers of integers, which influenced Abraham de Moivre's work later.

In the third part, Bernoulli applied the discussed probability techniques to the common chance games of the day—games played with cards or dice. He presented probability problems related to these and in addition, posed 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.

The fourth part discusses applying probability to civilibus, moralibus, and oeconomicis, or to personal, judicial, and financial decisions. In this section, Jakob differs from the school of thought known as frequentism, which defined probability in an empirical sense. He differed in a result resembling the law of large numbers, which Bernoulli described as predicting that the results of observation would approach theoretical probability as more trials were held, while the frequentists defined probability in terms of the former. Bernoulli was very proud of this result, referring to it as his "golden theorem", and remarked that it was "a problem in which I’ve engaged myself for twenty years". This early version of the law is known today as either Bernoulli's theorem or the weak law of large numbers, as it was less rigorous than the modern version.

Bernoulli appended to Ars Conjectandi a tract on calculus, which concerned infinite series. It was a reprint of five dissertations he had published between 1686 and 1704.

## Legacy

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

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. 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.

Thomas Simpson achieved a result that closely resembled de Moivre's. According to Simpsons' work's preface, his own work depended greatly on De Moivre's; De Moivre in fact described Simpson's work as an abridged version of his own. Thomas Bayes wrote an essay discussing theological implications of de Moivre's results. De Moivre's solution to a problem, namely that of determined the probability of an event by its relative frequency, was taken as a proof for the existence of God by Bayes.