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Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">1713 book on probability and combinatorics by Jacob Bernoulli</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arsconj.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Arsconj.gif/220px-Arsconj.gif" decoding="async" width="220" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Arsconj.gif/330px-Arsconj.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Arsconj.gif/440px-Arsconj.gif 2x" data-file-width="2908" data-file-height="2850" /></a><figcaption>The cover page of <span title="Latin-language text"><i lang="la">Ars Conjectandi</i></span></figcaption></figure> <p><b><span title="Latin-language text"><i lang="la">Ars Conjectandi</i></span></b> (<a href="/wiki/Latin" title="Latin">Latin</a> for "The Art of Conjecturing") is a book on <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and mathematical <a href="/wiki/Probability" title="Probability">probability</a> written by <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> and published in 1713, eight years after his death, by his nephew, <a href="/wiki/Nicolaus_I_Bernoulli" title="Nicolaus I Bernoulli">Niklaus Bernoulli</a>. The seminal work consolidated, apart from many combinatorial topics, many central ideas in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, such as the very first version of the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a>: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the <a href="/wiki/Twelvefold_way" title="Twelvefold way">twelvefold way</a> and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>. </p><p>Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>, <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>, <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>, and <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>. He incorporated fundamental combinatorial topics such as his theory of <a href="/wiki/Permutation" title="Permutation">permutations</a> and <a href="/wiki/Combination" title="Combination">combinations</a> (the aforementioned problems from the twelvefold way) as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a>, for instance. Core topics from probability, such as <a href="/wiki/Expected_value" title="Expected value">expected value</a>, were also a significant portion of this important work. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Christiaan-huygens4.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Christiaan-huygens4.jpg/200px-Christiaan-huygens4.jpg" decoding="async" width="200" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Christiaan-huygens4.jpg/300px-Christiaan-huygens4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Christiaan-huygens4.jpg/400px-Christiaan-huygens4.jpg 2x" data-file-width="1109" data-file-height="1154" /></a><figcaption>Christiaan Huygens published the first treatise on probability</figcaption></figure> <p>In Europe, the subject of <a href="/wiki/Probability" title="Probability">probability</a> was first formally developed in the 16th century with the work of <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>, whose interest in the branch of mathematics was largely due to his habit of gambling.<sup id="cite_ref-dunham_1-0" class="reference"><a href="#cite_note-dunham-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> He formalized what is now called the classical definition of probability: if an event has <i>a</i> possible outcomes and we select any <i>b</i> of those such that <i>b</i> ≤ <i>a</i>, the probability of any of the <i>b</i> occurring is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{smallmatrix}{\frac {b}{a}}\end{smallmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{smallmatrix}{\frac {b}{a}}\end{smallmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de5a1248b77a87c1018171a83099880bc8a66d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.294ex; height:2.843ex;" alt="{\displaystyle {\begin{smallmatrix}{\frac {b}{a}}\end{smallmatrix}}}"></span>. However, his actual influence on mathematical scene was not great; he wrote only one light tome on the subject titled <i>Liber de ludo aleae</i> (Book on Games of Chance), which was published posthumously in 1663.<sup id="cite_ref-second_2-0" class="reference"><a href="#cite_note-second-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mactutor_3-0" class="reference"><a href="#cite_note-mactutor-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The date which historians cite as the beginning of the development of modern probability theory is 1654, when two of the most well-known mathematicians of the time, Blaise Pascal and Pierre de Fermat, began a correspondence discussing the subject. The two initiated the communication because earlier that year, a gambler from <a href="/wiki/Paris" title="Paris">Paris</a> named <a href="/wiki/Antoine_Gombaud" title="Antoine Gombaud">Antoine Gombaud</a> had sent Pascal and other mathematicians several questions on the practical applications of some of these theories; in particular he posed the <a href="/wiki/Problem_of_points" title="Problem of points">problem of points</a>, concerning a theoretical two-player game in which a prize must be divided between the players due to external circumstances halting the game. The fruits of Pascal and Fermat's correspondence interested other mathematicians, including <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>, whose <i>De ratiociniis in aleae ludo</i> (Calculations in Games of Chance) appeared in 1657 as the final chapter of Van Schooten's <i>Exercitationes Matematicae</i>.<sup id="cite_ref-second_2-1" class="reference"><a href="#cite_note-second-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In 1665 Pascal posthumously published his results on the eponymous <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>, an important combinatorial concept. He referred to the triangle in his work <i>Traité du triangle arithmétique</i> (Traits of the Arithmetic Triangle) as the "arithmetic triangle".<sup id="cite_ref-britannica_4-0" class="reference"><a href="#cite_note-britannica-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In 1662, the book <i><a href="/wiki/Port-Royal_Logic" title="Port-Royal Logic">La Logique ou l’Art de Penser</a></i> was published anonymously in Paris.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The authors presumably were <a href="/wiki/Antoine_Arnauld" title="Antoine Arnauld">Antoine Arnauld</a> and <a href="/wiki/Pierre_Nicole" title="Pierre Nicole">Pierre Nicole</a>, two leading <a href="/wiki/Jansenists" class="mw-redirect" title="Jansenists">Jansenists</a>, who worked together with Blaise Pascal. The Latin title of this book is <i>Ars cogitandi</i>, which was a successful book on logic of the time. The <i>Ars cogitandi</i> consists of four books, with the fourth one dealing with decision-making under uncertainty by considering the analogy to gambling and introducing explicitly the concept of a quantified probability.<sup id="cite_ref-Collani2006_6-0" class="reference"><a href="#cite_note-Collani2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hacking1971_7-0" class="reference"><a href="#cite_note-Hacking1971-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>In the field of statistics and applied probability, <a href="/wiki/John_Graunt" title="John Graunt">John Graunt</a> published <i>Natural and Political Observations Made upon the Bills of Mortality</i> also in 1662, initiating the discipline of <a href="/wiki/Demography" title="Demography">demography</a>. This work, among other things, gave a statistical estimate of the population of London, produced the first life table, gave probabilities of survival of different age groups, examined the different causes of death, noting that the annual rate of suicide and accident is constant, and commented on the level and stability of sex ratio.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The usefulness and interpretation of Graunt's tables were discussed in a series of correspondences by brothers Ludwig and Christiaan Huygens in 1667, where they realized the difference between mean and median estimates and Christian even interpolated Graunt's life table by a smooth curve, creating the first continuous probability distribution; but their correspondences were not published. Later, <a href="/wiki/Johan_de_Witt" title="Johan de Witt">Johan de Witt</a>, the then prime minister of the Dutch Republic, published similar material in his 1671 work <i>Waerdye van Lyf-Renten</i> (A Treatise on Life Annuities), which used statistical concepts to determine <a href="/wiki/Life_expectancy" title="Life expectancy">life expectancy</a> for practical political purposes; a demonstration of the fact that this sapling branch of mathematics had significant pragmatic applications.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> De Witt's work was not widely distributed beyond the Dutch Republic, perhaps due to his fall from power and execution by mob in 1672. Apart from the practical contributions of these two work, they also exposed a fundamental idea that probability can be assigned to events that do not have inherent physical symmetry, such as the chances of dying at certain age, unlike say the rolling of a dice or flipping of a coin, simply by counting the frequency of occurrence. Thus, probability could be more than mere combinatorics.<sup id="cite_ref-Hacking1971_7-1" class="reference"><a href="#cite_note-Hacking1971-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Development_of_Ars_Conjectandi">Development of <i>Ars Conjectandi</i></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=2" title="Edit section: Development of Ars Conjectandi"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Jakob_Bernoulli.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Jakob_Bernoulli.jpg/220px-Jakob_Bernoulli.jpg" decoding="async" width="220" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Jakob_Bernoulli.jpg/330px-Jakob_Bernoulli.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Jakob_Bernoulli.jpg/440px-Jakob_Bernoulli.jpg 2x" data-file-width="2000" data-file-height="2338" /></a><figcaption>Portrait of Jakob Bernoulli in 1687</figcaption></figure> <p>In the wake of all these pioneers, Bernoulli produced many of the results contained in <i>Ars Conjectandi</i> between 1684 and 1689, which he recorded in his diary <i>Meditationes</i>.<sup id="cite_ref-dunham_1-1" class="reference"><a href="#cite_note-dunham-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> When he began the work in 1684 at the age of 30, while intrigued by combinatorial and probabilistic problems, Bernoulli had not yet read Pascal's work on the "arithmetic triangle" nor de Witt's work on the applications of probability theory: he had earlier requested a copy of the latter from his acquaintance <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a>, but Leibniz failed to provide it. The latter, however, did manage to provide Pascal's and Huygens' work, and thus it is largely upon these foundations that <i>Ars Conjectandi</i> is constructed.<sup id="cite_ref-shafer_11-0" class="reference"><a href="#cite_note-shafer-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Apart from these works, Bernoulli certainly possessed or at least knew the contents from secondary sources of the <i> La Logique ou l’Art de Penser</i> as well as Graunt's <i>Bills of Mortality</i>, as he makes explicit reference to these two works. </p><p>Bernoulli's progress over time can be pursued by means of the <i>Meditationes</i>. Three working periods with respect to his "discovery" can be distinguished by aims and times. The first period, which lasts from 1684 to 1685, is devoted to the study of the problems regarding the games of chance posed by Christiaan Huygens; during the second period (1685-1686) the investigations are extended to cover processes where the probabilities are not known a priori, but have to be determined a posteriori. Finally, in the last period (1687-1689), the problem of measuring the probabilities is solved.<sup id="cite_ref-Collani2006_6-1" class="reference"><a href="#cite_note-Collani2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Before the publication of his <i>Ars Conjectandi</i>, Bernoulli had produced a number of treatises related to probability:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <ul><li><i>Parallelismus ratiocinii logici et algebraici</i>, Basel, 1685.</li> <li>In the <i>Journal des Sçavans</i> 1685 (26.VIII), p. 314 there appear two problems concerning the probability each of two players may have of winning in a game of dice. Solutions were published in the <i>Acta Eruditorum</i> 1690 (May), pp. 219–223 in the article <i>Quaestiones nonnullae de usuris, cum solutione Problematis de Sorte Alearum</i>. In addition, Leibniz himself published a solution in the same journal on pages 387-390.</li> <li><i>Theses logicae de conversione et oppositione enunciationum</i>, a public lecture delivered at Basel, 12 February 1686. Theses XXXI to XL are related to the theory of probability.</li> <li><i>De Arte Combinatoria Oratio Inauguralis</i>, 1692.</li> <li>The Letter <i>à un amy sur les parties du jeu de paume</i>, that is, a letter to a friend on sets in the game of Tennis, published with the Ars Conjectandi in 1713.</li></ul> <p>Between 1703 and 1705, Leibniz corresponded with Jakob after learning about his discoveries in probability from his brother <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann</a>.<sup id="cite_ref-Sylla_13-0" class="reference"><a href="#cite_note-Sylla-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Leibniz managed to provide thoughtful criticisms on Bernoulli's law of large numbers, but failed to provide Bernoulli with de Witt's work on annuities that he so desired.<sup id="cite_ref-Sylla_13-1" class="reference"><a href="#cite_note-Sylla-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> From the outset, Bernoulli wished for his work to demonstrate that combinatorics and probability theory would have numerous real-world applications in all facets of society—in the line of Graunt's and de Witt's work— and would serve as a rigorous method of logical reasoning under insufficient evidence, as used in courtrooms and in moral judgements. It was also hoped that the theory of probability could provide comprehensive and consistent method of reasoning, where ordinary reasoning might be overwhelmed by the complexity of the situation.<sup id="cite_ref-Sylla_13-2" class="reference"><a href="#cite_note-Sylla-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Thus the title <i>Ars Conjectandi</i> was chosen: a link to the concept of <i><a href="/wiki/Ars_inveniendi" class="mw-redirect" title="Ars inveniendi">ars inveniendi</a></i> from <a href="/wiki/Scholasticism" title="Scholasticism">scholasticism</a>, which provided the symbolic link to pragmatism he desired and also as an extension of the prior <i>Ars Cogitandi</i>.<sup id="cite_ref-Collani2006_6-2" class="reference"><a href="#cite_note-Collani2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In Bernoulli's own words, the "art of conjecture" is defined in Chapter II of Part IV of his <i>Ars Conjectandi</i> as: </p> <blockquote> <p>The art of measuring, as precisely as possible, probabilities of things, with the goal that we would be able always to choose or follow in our judgments and actions that course, which will have been determined to be better, more satisfactory, safer or more advantageous. </p> </blockquote> <p>The development of the book was terminated by Bernoulli's death in 1705; thus the book is essentially incomplete when compared with Bernoulli's original vision. The quarrel with his younger brother Johann, who was the most competent person who could have fulfilled Jacob's project, prevented Johann to get hold of the manuscript. Jacob's own children were not mathematicians and were not up to the task of editing and publishing the manuscript. Finally Jacob's nephew Niklaus, 7 years after Jacob's death in 1705, managed to publish the manuscript in 1713.<sup id="cite_ref-bernoulli_14-0" class="reference"><a href="#cite_note-bernoulli-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Contents">Contents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=3" title="Edit section: Contents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:JakobBernoulliSummaePotestatum.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/220px-JakobBernoulliSummaePotestatum.png" decoding="async" width="220" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/330px-JakobBernoulliSummaePotestatum.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/440px-JakobBernoulliSummaePotestatum.png 2x" data-file-width="576" data-file-height="732" /></a><figcaption>Cutout of a page from <i>Ars Conjectandi</i> showing Bernoulli's formula for sum of integer powers. The last line gives his eponymous numbers.</figcaption></figure> <p>Bernoulli's work, originally published in Latin<sup id="cite_ref-schneider_16-0" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> is divided into four parts.<sup id="cite_ref-shafer_11-1" class="reference"><a href="#cite_note-shafer-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It covers most notably his theory of permutations and combinations; the standard foundations of combinatorics today and subsets of the foundational problems today known as the <a href="/wiki/Twelvefold_way" title="Twelvefold way">twelvefold way</a>. It also discusses the motivation and applications of a sequence of numbers more closely related to <a href="/wiki/Number_theory" title="Number theory">number theory</a> than probability; these <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a> bear his name today, and are one of his more notable achievements.<sup id="cite_ref-britannica2_17-0" class="reference"><a href="#cite_note-britannica2-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-columbia_18-0" class="reference"><a href="#cite_note-columbia-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The first part is an in-depth expository on Huygens' <i>De ratiociniis in aleae ludo</i>. Bernoulli provides in this section solutions to the five problems Huygens posed at the end of his work.<sup id="cite_ref-shafer_11-2" class="reference"><a href="#cite_note-shafer-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> He particularly develops Huygens' concept of expected value—the weighted average of all possible outcomes of an event. Huygens had developed the following formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {p_{0}a_{0}+p_{1}a_{1}+p_{2}a_{2}+\cdots +p_{n}a_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {p_{0}a_{0}+p_{1}a_{1}+p_{2}a_{2}+\cdots +p_{n}a_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/396e21bf71c233be1e6c68d0126f0263e8573d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.801ex; height:5.676ex;" alt="{\displaystyle E={\frac {p_{0}a_{0}+p_{1}a_{1}+p_{2}a_{2}+\cdots +p_{n}a_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}.}"></span><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></dd></dl> <p>In this formula, <i>E</i> is the expected value, <i>p<sub>i</sub></i> are the probabilities of attaining each value, and <i>a<sub>i</sub></i> are the attainable values. Bernoulli normalizes the expected value by assuming that <i>p<sub>i</sub></i> are the probabilities of all the disjoint outcomes of the value, hence implying that <i>p</i><sub>0</sub> + <i>p</i><sub>1</sub> + ... + <i>p</i><sub><i>n</i></sub> = 1. Another key theory developed in this part is the probability of achieving at least a certain number of successes from a number of binary events, today named <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trials</a>,<sup id="cite_ref-dunhamtmu_20-0" class="reference"><a href="#cite_note-dunhamtmu-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> given that the probability of success in each event was the same. Bernoulli shows through <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a> that given <i>a</i> the number of favorable outcomes in each event, <i>b</i> the number of total outcomes in each event, <i>d</i> the desired number of successful outcomes, and <i>e</i> the number of events, the probability of at least <i>d</i> successes is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\sum _{i=0}^{e-d}{\binom {e}{d+i}}\left({\frac {a}{b}}\right)^{d+i}\left({\frac {b-a}{b}}\right)^{e-d-i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>−</mo> <mi>d</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>e</mi> <mrow> <mi>d</mi> <mo>+</mo> <mi>i</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>+</mo> <mi>i</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>−</mo> <mi>d</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\sum _{i=0}^{e-d}{\binom {e}{d+i}}\left({\frac {a}{b}}\right)^{d+i}\left({\frac {b-a}{b}}\right)^{e-d-i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5977215c6b6a4da100a9369e2cd276ff5c769fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.6ex; height:7.343ex;" alt="{\displaystyle P=\sum _{i=0}^{e-d}{\binom {e}{d+i}}\left({\frac {a}{b}}\right)^{d+i}\left({\frac {b-a}{b}}\right)^{e-d-i}.}"></span><sup id="cite_ref-schneider2_21-0" class="reference"><a href="#cite_note-schneider2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The first part concludes with what is now known as the <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a>.<sup id="cite_ref-schneider_16-1" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The second part expands on enumerative combinatorics, or the systematic numeration of objects. It was in this part that two of the most important of the twelvefold ways—the permutations and combinations that would form the basis of the subject—were fleshed out, though they had been introduced earlier for the purposes of probability theory. He gives the first non-inductive proof of the binomial expansion for integer exponent using combinatorial arguments. On a note more distantly related to combinatorics, the second section also discusses the general formula for sums of integer powers; the free coefficients of this formula are therefore called the <a href="/wiki/Bernoulli_numbers" class="mw-redirect" title="Bernoulli numbers">Bernoulli numbers</a>, which influenced Abraham de Moivre's work later,<sup id="cite_ref-schneider_16-2" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and which have proven to have numerous applications in number theory.<sup id="cite_ref-maseres_22-0" class="reference"><a href="#cite_note-maseres-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In the third part, Bernoulli applies the probability techniques from the first section to the common chance games played with playing cards or dice.<sup id="cite_ref-shafer_11-3" class="reference"><a href="#cite_note-shafer-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> He does not feel the necessity to describe the rules and objectives of the card games he analyzes. He presents probability problems related to these games and, once a method had been established, posed generalizations. For example, a problem involving the expected number of "court cards"—jack, queen, and king—one would pick in a five-card hand from a standard deck of 52 cards containing 12 court cards could be generalized to a deck with <i>a</i> cards that contained <i>b</i> court cards, and a <i>c</i>-card hand.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The fourth section continues the trend of practical applications by discussing applications of probability to <i>civilibus</i>, <i>moralibus</i>, and <i>oeconomicis</i>, or to personal, judicial, and financial decisions. In this section, Bernoulli differs from the school of thought known as <a href="/wiki/Frequency_probability" class="mw-redirect" title="Frequency probability">frequentism</a>, which defined probability in an empirical sense.<sup id="cite_ref-shafer18_24-0" class="reference"><a href="#cite_note-shafer18-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> As a counter, he produces a result resembling the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a>, which he describes as predicting that the results of observation would approach theoretical probability as more trials were held—in contrast, frequents <i>defined</i> probability in terms of the former.<sup id="cite_ref-bernoulli_14-1" class="reference"><a href="#cite_note-bernoulli-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Bernoulli was very proud of this result, referring to it as his "golden theorem",<sup id="cite_ref-dunhamtmu2_25-0" class="reference"><a href="#cite_note-dunhamtmu2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and remarked that it was "a problem in which I've engaged myself for twenty years".<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> This early version of the law is known today as either Bernoulli's theorem or the weak law of large numbers, as it is less rigorous and general than the modern version.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>After these four primary expository sections, almost as an afterthought, Bernoulli appended to <i>Ars Conjectandi</i> a tract on <a href="/wiki/Calculus" title="Calculus">calculus</a>, which concerned <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>.<sup id="cite_ref-schneider_16-3" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> It was a reprint of five dissertations he had published between 1686 and 1704.<sup id="cite_ref-schneider2_21-1" class="reference"><a href="#cite_note-schneider2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Legacy">Legacy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=4" title="Edit section: Legacy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Abraham_de_moivre.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Abraham_de_moivre.jpg/220px-Abraham_de_moivre.jpg" decoding="async" width="220" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Abraham_de_moivre.jpg/330px-Abraham_de_moivre.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Abraham_de_moivre.jpg/440px-Abraham_de_moivre.jpg 2x" data-file-width="523" data-file-height="666" /></a><figcaption>Abraham de Moivre's work was built in part on Bernoulli's</figcaption></figure> <p><i>Ars Conjectandi</i> is considered a landmark work in combinatorics and the founding work of mathematical probability.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Among others, an anthology of great mathematical writings published by <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a> and edited by historian <a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Ivor Grattan-Guinness</a> describes the studies set out in the work "[occupying] mathematicians throughout 18th and 19th centuries"—an influence lasting three centuries.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Statistician <a href="/wiki/A._W._F._Edwards" title="A. W. F. Edwards">Anthony Edwards</a> praised not only the book's groundbreaking content, writing that it demonstrated Bernoulli's "thorough familiarity with the many facets [of combinatorics]," but its form: "[Ars Conjectandi] is a very well-written book, excellently constructed."<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Perhaps most recently, notable popular mathematical historian and topologist William Dunham called the paper "the next milestone of probability theory [after the work of Cardano]" as well as "Jakob Bernoulli's masterpiece".<sup id="cite_ref-dunham_1-2" class="reference"><a href="#cite_note-dunham-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It greatly aided what Dunham describes as "Bernoulli's long-established reputation".<sup id="cite_ref-dunham2_33-0" class="reference"><a href="#cite_note-dunham2-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>Bernoulli's work influenced many contemporary and subsequent mathematicians. Even the afterthought-like tract on calculus has been quoted frequently; most notably by the Scottish mathematician <a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a>.<sup id="cite_ref-schneider_16-4" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Jacob's program of applying his art of conjecture to the matters of practical life, which was terminated by his death in 1705, was continued by his nephew <a href="/wiki/Nicolaus_I_Bernoulli" title="Nicolaus I Bernoulli">Nicolaus Bernoulli</a>, after having taken parts verbatim out of <i>Ars Conjectandi</i>, for his own dissertation entitled <i>De Usu Artis Conjectandi in Jure</i> which was published already in 1709.<sup id="cite_ref-Collani2006_6-3" class="reference"><a href="#cite_note-Collani2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Nicolas finally edited and assisted in the publication of <i>Ars conjectandi</i> in 1713. Later Nicolaus also edited Jacob Bernoulli's complete works and supplemented it with results taken from Jacob's diary.<sup id="cite_ref-MacTutor-Nicolas_34-0" class="reference"><a href="#cite_note-MacTutor-Nicolas-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Pierre_R%C3%A9mond_de_Montmort" class="mw-redirect" title="Pierre Rémond de Montmort">Pierre Rémond de Montmort</a>, in collaboration with Nicolaus Bernoulli, wrote a book on probability <i><a href="/wiki/Essay_d%27analyse_sur_les_jeux_de_hazard" title="Essay d'analyse sur les jeux de hazard">Essay d'analyse sur les jeux de hazard</a></i> which appeared in 1708, which can be seen as an extension of the Part III of <i>Ars Conjectandi</i> which applies combinatorics and probability to analyze games of chance commonly played at that time.<sup id="cite_ref-MacTutor-Nicolas_34-1" class="reference"><a href="#cite_note-MacTutor-Nicolas-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> also wrote extensively on the subject in <i>De mensura sortis: Seu de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus</i> of 1711 and its extension <i><a href="/wiki/The_Doctrine_of_Chances" title="The Doctrine of Chances"> The Doctrine of Chances or, a Method of Calculating the Probability of Events in Play</a></i> of 1718.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> De Moivre's most notable achievement in probability was the discovery of the first instance of <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, by which he was able to approximate the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> with the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>.<sup id="cite_ref-schneider_16-5" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> To achieve this De Moivre developed an <a href="/wiki/Asymptote" title="Asymptote">asymptotic</a> sequence for the <a href="/wiki/Factorial" title="Factorial">factorial</a> function —- which we now refer to as <a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a> —- and Bernoulli's formula for the sum of powers of numbers.<sup id="cite_ref-schneider_16-6" class="reference"><a href="#cite_note-schneider-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Both Montmort and de Moivre adopted the term <i>probability</i> from Jacob Bernoulli, which had not been used in all the previous publications on gambling, and both their works were enormously popular.<sup id="cite_ref-Collani2006_6-4" class="reference"><a href="#cite_note-Collani2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The refinement of Bernoulli's Golden Theorem, regarding the convergence of theoretical probability and empirical probability, was taken up by many notable latter day mathematicians like De Moivre, Laplace, Poisson, Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. The complete proof of the Law of Large Numbers for the arbitrary random variables was finally provided during first half of 20th century.<sup id="cite_ref-FOOTNOTESeneta2013_36-0" class="reference"><a href="#cite_note-FOOTNOTESeneta2013-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>A significant indirect influence was <a href="/wiki/Thomas_Simpson" title="Thomas Simpson">Thomas Simpson</a>, who achieved a result that closely resembled de Moivre's. According to Simpsons' work's preface, his own work depended greatly on de Moivre's; the latter in fact described Simpson's work as an abridged version of his own.<sup id="cite_ref-schneider11_37-0" class="reference"><a href="#cite_note-schneider11-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Finally, <a href="/wiki/Thomas_Bayes" title="Thomas Bayes">Thomas Bayes</a> wrote an essay discussing <a href="/wiki/Theology" title="Theology">theological</a> implications of de Moivre's results: his solution to a problem, namely that of determining the probability of an event by its relative frequency, was taken as a proof for the <a href="/wiki/Existence_of_God" title="Existence of God">existence of God</a> by Bayes.<sup id="cite_ref-schneider14_38-0" class="reference"><a href="#cite_note-schneider14-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Finally in 1812, <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> published his <i>Théorie analytique des probabilités</i> in which he consolidated and laid down many fundamental results in probability and statistics such as the moment generating function, method of least squares, inductive probability, and hypothesis testing, thus completing the final phase in the development of classical probability. Indeed, in light of all this, there is good reason Bernoulli's work is hailed as such a seminal event; not only did his various influences, direct and indirect, set the mathematical study of combinatorics spinning, but even theology was impacted. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial distribution</a></li> <li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li> <li><a href="/wiki/Law_of_Large_Numbers" class="mw-redirect" title="Law of Large Numbers">Law of Large Numbers</a></li> <li><a href="/wiki/Bernoulli_Numbers" class="mw-redirect" title="Bernoulli Numbers">Bernoulli Numbers</a></li> <li><a href="/wiki/Binomial_Distribution" class="mw-redirect" title="Binomial Distribution">Binomial Distribution</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-dunham-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-dunham_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dunham_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-dunham_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDunham1990">Dunham 1990</a>, p. 191</span> </li> <li id="cite_note-second-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-second_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-second_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output 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class="reference-text"><a href="#CITEREFBrakel1976">Brakel 1976</a>, p. 123</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFShafer1996">Shafer 1996</a></span> </li> <li id="cite_note-shafer-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-shafer_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-shafer_11-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-shafer_11-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-shafer_11-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFShafer1996">Shafer 1996</a>, pp. 3–4</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPulskamp" class="citation cs2">Pulskamp, Richard J., <a rel="nofollow" class="external text" href="http://cerebro.xu.edu/math/Sources/JakobBernoulli/JakobB.html"><i>Jakob Bernoulli</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">1 March</span> 2013</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Jakob+Bernoulli&rft.aulast=Pulskamp&rft.aufirst=Richard+J.&rft_id=http%3A%2F%2Fcerebro.xu.edu%2Fmath%2FSources%2FJakobBernoulli%2FJakobB.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></span> </li> <li id="cite_note-Sylla-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sylla_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sylla_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Sylla_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSylla1998">Sylla 1998</a></span> </li> <li id="cite_note-bernoulli-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-bernoulli_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bernoulli_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBernoulli2005">Bernoulli 2005</a>, p. i</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation cs2">Weisstein, Eric, <a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/biography/BernoulliJakob.html"><i>Bernoulli, Jakob</i></a>, Wolfram<span class="reference-accessdate">, retrieved <span class="nowrap">2008-06-09</span></span></cite><span 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minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>r</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{smallmatrix}{\binom {n}{r}}\end{smallmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8746bfef1850892b85ec57777b6d7d1aa877ee03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.515ex; height:3.509ex;" alt="{\displaystyle {\begin{smallmatrix}{\binom {n}{r}}\end{smallmatrix}}}"></span> represents the number of ways to choose <i>r</i> objects from a set of <i>n</i> distinguishable objects without replacement.</span> </li> <li id="cite_note-dunhamtmu-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-dunhamtmu_20-0">^</a></b></span> <span 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href="#CITEREFShafer1996">Shafer 1996</a>, pp. 18</span> </li> <li id="cite_note-dunhamtmu2-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-dunhamtmu2_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDunham1994">Dunham 1994</a>, pp. 17–18</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolasek2000" class="citation cs2">Polasek, Wolfgang (August 2000), "The Bernoullis and the Origin of Probability Theory", <i>Resonance</i>, vol. 26, no. 42, Indian Academy of Sciences</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Resonance&rft.atitle=The+Bernoullis+and+the+Origin+of+Probability+Theory&rft.volume=26&rft.issue=42&rft.date=2000-08&rft.aulast=Polasek&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Weak_Law_of_Large_Numbers"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WeakLawofLargeNumbers.html">"Weak Law of Large Numbers"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Weak+Law+of+Large+Numbers&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FWeakLawofLargeNumbers.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><a href="#CITEREFBernoulli2005">Bernoulli 2005</a>. Preface by Sylla, vii.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><a href="#CITEREFHald2005">Hald 2005</a>, p. 253</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><a href="#CITEREFMaĭstrov1974">Maĭstrov 1974</a>, p. 66</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><a href="#CITEREFElsevier2005">Elsevier 2005</a>, p. 103</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><a href="#CITEREFEdwards1987">Edwards 1987</a>, p. 154</span> </li> <li id="cite_note-dunham2-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-dunham2_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDunham1990">Dunham 1990</a>, p. 192</span> </li> <li id="cite_note-MacTutor-Nicolas-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-MacTutor-Nicolas_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MacTutor-Nicolas_34-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(I).html">"Nicolaus(I) Bernoulli"</a>. The MacTutor History of Mathematics Archive<span class="reference-accessdate">. Retrieved <span class="nowrap">22 Aug</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Nicolaus%28I%29+Bernoulli&rft.pub=The+MacTutor+History+of+Mathematics+Archive&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FBiographies%2FBernoulli_Nicolaus%28I%29.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><a href="#CITEREFde_Moivre2000">de Moivre 2000</a>, p. i</span> </li> <li id="cite_note-FOOTNOTESeneta2013-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESeneta2013_36-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSeneta2013">Seneta 2013</a>.</span> </li> <li id="cite_note-schneider11-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-schneider11_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchneider2006">Schneider 2006</a>, p. 11</span> </li> <li id="cite_note-schneider14-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-schneider14_38-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchneider2006">Schneider 2006</a>, p. 14</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli,_Jakob1713" class="citation cs2">Bernoulli, Jakob (1713), <i>Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis</i>, Basel: Thurneysen Brothers, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/7073795">7073795</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ars+conjectandi%2C+opus+posthumum.+Accedit+Tractatus+de+seriebus+infinitis%2C+et+epistola+gallic%C3%A9+scripta+de+ludo+pilae+reticularis&rft.place=Basel&rft.pub=Thurneysen+Brothers&rft.date=1713&rft_id=info%3Aoclcnum%2F7073795&rft.au=Bernoulli%2C+Jakob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli2005" class="citation cs2">Bernoulli, Jakob (2005) [1713], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-xgwSAjTh34C&q=edith+dudley+sylla"><i>The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis (English translation)</i></a>, translated by Edith Sylla, Baltimore: Johns Hopkins Univ Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-8235-7" title="Special:BookSources/978-0-8018-8235-7"><bdi>978-0-8018-8235-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Conjecturing%2C+together+with+Letter+to+a+Friend+on+Sets+in+Court+Tennis+%28English+translation%29&rft.place=Baltimore&rft.pub=Johns+Hopkins+Univ+Press&rft.date=2005&rft.isbn=978-0-8018-8235-7&rft.aulast=Bernoulli&rft.aufirst=Jakob&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-xgwSAjTh34C%26q%3Dedith%2Bdudley%2Bsylla&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli,_Jakob2005" class="citation cs2">Bernoulli, Jakob (2005) [1713], <a rel="nofollow" class="external text" href="http://www.sheynin.de/download/bernoulli.pdf"><i>On the Law of Large Numbers, Part Four of Ars Conjectandi (English translation)</i></a> <span class="cs1-format">(PDF)</span>, translated by Oscar Sheynin, Berlin: NG Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-938417-14-0" title="Special:BookSources/978-3-938417-14-0"><bdi>978-3-938417-14-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Law+of+Large+Numbers%2C+Part+Four+of+Ars+Conjectandi+%28English+translation%29&rft.place=Berlin&rft.pub=NG+Verlag&rft.date=2005&rft.isbn=978-3-938417-14-0&rft.au=Bernoulli%2C+Jakob&rft_id=http%3A%2F%2Fwww.sheynin.de%2Fdownload%2Fbernoulli.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli2002" class="citation cs2">Bernoulli, Jakob (2002) [1713], <a rel="nofollow" class="external text" href="http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABZ9501"><i>Wahrscheinlichkeitsrechnung (Ars conjectandi) (German translation)</i></a>, translated by Haussner, Robert, Frankfurt am Main: <a href="/wiki/Verlag_Harri_Deutsch" title="Verlag Harri Deutsch">Verlag Harri Deutsch</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-8171-3107-5" title="Special:BookSources/978-3-8171-3107-5"><bdi>978-3-8171-3107-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wahrscheinlichkeitsrechnung+%28Ars+conjectandi%29+%28German+translation%29&rft.place=Frankfurt+am+Main&rft.pub=Verlag+Harri+Deutsch&rft.date=2002&rft.isbn=978-3-8171-3107-5&rft.aulast=Bernoulli&rft.aufirst=Jakob&rft_id=http%3A%2F%2Fquod.lib.umich.edu%2Fcgi%2Ft%2Ftext%2Ftext-idx%3Fc%3Dumhistmath%3Bidno%3DABZ9501&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrakel1976" class="citation cs2">Brakel, J. van (June 1976), "Some Remarks on the Prehistory of the Concept of Statistical Probability", <i>Archive for History of Exact Sciences</i>, <b>16</b> (2): 119, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00349634">10.1007/BF00349634</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119997834">119997834</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Some+Remarks+on+the+Prehistory+of+the+Concept+of+Statistical+Probability&rft.volume=16&rft.issue=2&rft.pages=119&rft.date=1976-06&rft_id=info%3Adoi%2F10.1007%2FBF00349634&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119997834%23id-name%3DS2CID&rft.aulast=Brakel&rft.aufirst=J.+van&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" 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href="http://www.emis.de/journals/JEHPS/Juin2006/Schneider.pdf">"Direct and Indirect Influences of Jakob Bernoulli's Ars Conjectandi in 18th Century Great Britain"</a> <span class="cs1-format">(PDF)</span>, <i>Electronic Journal for the History of Probability and Statistics</i>, vol. 2, no. 1</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+for+the+History+of+Probability+and+Statistics&rft.atitle=Direct+and+Indirect+Influences+of+Jakob+Bernoulli%27s+Ars+Conjectandi+in+18th+Century+Great+Britain&rft.volume=2&rft.issue=1&rft.date=2006-06&rft.aulast=Schneider&rft.aufirst=Ivo&rft_id=http%3A%2F%2Fwww.emis.de%2Fjournals%2FJEHPS%2FJuin2006%2FSchneider.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchneider1984" class="citation cs2">Schneider, Ivo (1984), "The 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Econometrics&rft.atitle=The+Significance+of+Jacob+Bernoulli%27s+Ars+Conjectandi+for+the+Philosophy+of+Probability+Today&rft.volume=75&rft.issue=1&rft.pages=15-32&rft.date=1996&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.407.1066%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1016%2F0304-4076%2895%2901766-6&rft.aulast=Shafer&rft.aufirst=Glenn&rft_id=http%3A%2F%2Fwww.glennshafer.com%2Fassets%2Fdownloads%2Farticles%2Farticle55.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSheynin1968" class="citation cs2">Sheynin, O.B. (Nov 1968), "Studies in the History of Probability and Statistics. XXI.: On the Early History of the Law of Large Numbers", <i>Biometrika</i>, <b>55</b> (3): 459–467, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2334251">10.2307/2334251</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2334251">2334251</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Studies+in+the+History+of+Probability+and+Statistics.+XXI.%3A+On+the+Early+History+of+the+Law+of+Large+Numbers&rft.volume=55&rft.issue=3&rft.pages=459-467&rft.date=1968-11&rft_id=info%3Adoi%2F10.2307%2F2334251&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2334251%23id-name%3DJSTOR&rft.aulast=Sheynin&rft.aufirst=O.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSylla1998" class="citation cs2">Sylla, Edith D. (1998), <a rel="nofollow" class="external text" href="http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1sylla.html">"The Emergence of Mathematical Probability from the Perspective of the Leibniz-Jacob Bernoulli Correspondence"</a>, <i>Perspectives on Science</i>, <b>6</b> (1&2): 41–76</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Perspectives+on+Science&rft.atitle=The+Emergence+of+Mathematical+Probability+from+the+Perspective+of+the+Leibniz-Jacob+Bernoulli+Correspondence&rft.volume=6&rft.issue=1%262&rft.pages=41-76&rft.date=1998&rft.aulast=Sylla&rft.aufirst=Edith+D.&rft_id=http%3A%2F%2Fmuse.jhu.edu%2Fjournals%2Fperspectives_on_science%2Fv006%2F6.1sylla.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTodhunter1865" class="citation cs2">Todhunter, Isaac (1865), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=04ARAAAAYAAJ"><i>A History of the Mathematical Theory of Probability from the time of Pascal to that of Laplace</i></a>, Macmillan and Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Mathematical+Theory+of+Probability+from+the+time+of+Pascal+to+that+of+Laplace&rft.pub=Macmillan+and+Co.&rft.date=1865&rft.aulast=Todhunter&rft.aufirst=Isaac&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D04ARAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArs+Conjectandi" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ars_Conjectandi&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Quotations/Bernoulli_Jacob.html">Quotations by Jakob Bernoulli</a></li> <li><a rel="nofollow" class="external text" href="http://cerebro.xu.edu/math/Sources/index.html">Sources in the History of Probability and Statistics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130513022626/http://cerebro.xu.edu/math/Sources/index.html">Archived</a> 2013-05-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130512180959/http://statprob.com/encyclopedia/JakobBERNOULLI.html">Biography of Jakob Bernoulli</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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