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The Doctrine of Chances

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The Doctrine of Chances N DOCTRINETHE CHANCES:O F O R, A Method of Calculating the Probabilities of Events in Play. THE THIRD EDITION, Fuller, Clearer, and more CerreSl than the Former. By A. D E M O I V R E, Fellow of the Royal Society, and Member of the Royal Academies OF Sciences of Berlin and Paris. LONDON: Printed for A. Millar, in the S^ra^d. MDCCLVI, |q n D m Q n a u a u q| To the Right Honourable the Lord CARP ENTER.* MY LORD, HERE are many People in the World who arc prepoffefled with an Opinion, that the Dodrine of Chances has a Ten- dency to promote Play; but they foon will be undeceived, if they think fit to look into the general Defign of this Book : in the mean time it will not be improper to in- form them, that your Lordfhip is pleafed to efpoufe the Patronage of this fecond Edition; which your ftiidl Pro- bity, and the diftinguifhed Charader you bear in the World, would not have permitted, were not theii' Ap- prehenfions altogether groundlefs, * Tliis Dedication was prefixed to the 2d Edition. Your 880G';'li DEDICATION. Your Lordfhip does eafily perceive, that this Dodlrine is fo far from encouraging Play, that it is rather a Guard againft it, by fctting in a clear Light, the Advantages and Difad vantages ofthofe Games wherein Chance is concerned. Befides the Endowments of the Mind which you have in common with Thofe w^hofe natural Talents have been cultivated by the beft Education, you have this particular Happinefs, that you underftand, in an eminent Degree, the Principles of Political Arithmetic, the Nature of our Funds, the National Credit, and its Influence on public Affairs. As one Branch of this ufeful Knowledge extends to the Valuation of Annuities founded on the Contingencies of Life, and that I have made it my particular Care to facili- tate and improve the Rules I have formerly given on that Subjeft ; I flatter mylclf with a favourable Acceptance of what is now, with the gr^atefl: Deference, fubmitted to your Judgment, by, My Lord, Your Lord/hlfs Moft Obedient and Mojl Obliged, Humble Servant, A. de Moivre. %I# %I# -;>-^<j- J^ %>:#-"a^cf* %*^*a*d?* ns'^tf' «^*ir» ~i-^tf' %*a^-a^-cT' %*# Iofa (*ocJoo&oi!S3o*c5ooSoo5ooto PREFACE 5F~»~^75 wo TO ^^flw/ ^^ip^w 2>^n, fince I gave a SpecimeJi in the I Philofophical Tranladtions, of what I now more largely I treat of in this Book. I'be occafion of my then undertaking -^^ this SubjeSi was chiejly oicitig to the Defire and Rncourage- ment the Honourable Francis Robartes ivho, upon occapon of 'f- Efq; of a French Tra5i, called, L'Analyl'e des Jcux de Hazard, ivhich had lately been publijhed^ ivas pleafed to propofe to me fame Problems of much greater difficulty than any he had Jound in that Book ; which having fohed to his SatisfaStion, he engaged me to methodize thofe Problems, and to lay down the Rules which had led me to their Solution. After I had pro- ceeded thus far, it was enjoined me by the Royal Society, to communicate to them what I had difcovered on this SubjeSl : afid thereupon it was or- dered to be publijhed in the TranJaSiiotis, not fo much as a matter relating to Play, but as containing fome gefieral Speculations not un-xorthy to be confidered by the hovers of Truth. I had not at that time read any thing concerning this SubjcSi, but Mr. Huygen'i Book de Ratiociniis in Ludo Ales, atjd a Utile E?igliJ}} Piece (which was properly a Tranfation oj it) done by a very ingenious Gentleman, who, tho' capable of carrying the tnatter a great dealfar- ther, was contented to follow his Original ; adding only to it the com- putation of the Advajjtage of the Setter in the Play called Hazard, and fome few things more. As for the French Book, I had run it over but curforily, by reafon I had obferved that the Author chiefly injijied on * This Preface was wrttttn in 1717. + Now Earl o/KADtiOR. the ii PREFACE. the Method cf Huygens, ivhich I was abfolutely rcfohed to rejeSi^ as not feeming to me to be the geriuine and natural ivay ^f coming at the So- lution of Prcblems of this kind. I had [aid in my Specimen, that Mr. Huygens iioas the Jirft who hadpublifjed the Rules of this Calculation, i>2tending thereby io do jufiice to a Man ivho had ivell deferved cf the Public; but what I then faid lias mi/ivterpreted, as if I had dejigned to urong fome Perfons who bad ccnfdered this matter before him : and a Pafjage was cited againjl me out of Huygcn'i Preface, in which hefaith. Sciendum vero quod iam pridem, inter Praeftantiflimos tota Gallia Geometras, Calculus hie fuerit agitatus ; ne quis indebitam mihi primae Invcntionis glo- riani hac in re tribuat. But what follows immediately after, had it been minded, might have cleared me from any Sufpicion oj injuftice. 'The words are thefe, Casterum illi difficillimis quibufque Quseftionibus fe invicem exercere foliti, methodum fuam quifque occultam reti- nuere, adeo ut a prim is dementis hanc materiam evolvere mihi necefie fuerit. By which it appears, that tho' Mr. Huygens was not the fifl who had applied himfelfto tkofe forts of ^efiions, he was ne- "jerthelefs the firfi who had publified Rules for their Solution ; which is all that I ajirmed. ^ Such a 7ra6l as this is may be ufeful to feveral ends; the firfl of which is, that there being in the World feveral inquifitive Perfons, who are defirous to know what foundation they go upon, when they engage in Play, whether from a motive of Gain, or barely Diver- jion, thy may, by the help of this or the like TraB, gratijy their CU' riofity, either by taking the pains to underftand what is here Demofi- ftrated, or el/e making ife of the Conclufions, and taking it for granted that the Demonftrations are right. Another ufe to be made of this DoSirim of Chances is, that it may Jerve i): Conjun^ion with the other farts oJ the Mathematicks, as a ft Introduction to the Art of Rcafoning; it being known by experience that nothing can contribute more to the attaining of that Art, than the con[deration of a long Train of Confequences, rightly deduced from undoubted Principles, of which this Book affords many Examples. To this may be added, that fome of the Problems about Chance having a great appearance of Simplicity, the Mind is eafdy drawn into a belief, that their Solution may be attained by the meer Strength of natural good Se?fe; which generally proving otherwife, and the Mif- takes occafioned thereby being not uifrequent, 'tis prefumed that a Book of this Kind, which teaches to djftinguifh Truth from what feems help to good Rea- fo nearly to refemble it, will be looked upon as a Jbning. Among PREFACE. iii Among the fe^oeral Mijlakes that are committed about Chance, one of the mojl common and leaft JufpeSied^ is that "which relates to Lotteries. ^hus, Juppofing a Lottery wherein the proportion of the Blanks to the Prizes is as fiiie to one ; 'tis very natural to conclude, that therefore five Tickets are requifite for the Chance of a Prize ; and yet it may be proved, Demofijlratively, thatfour Tickets are more than Jufficient for that pur- pofe, -which will be confirmed by often repeated Experience. In the like manner, fuppofmg a Lottery nvherein the proportion of the Blanks to the Prizes is as Thirty -nine to One, (fuch as was the L',ttery of 1710^ it may be proved, that in twenty eight Tickets, a Prize is as likely to be taken as mt ; iihich tho' it mayfeem to contradiB the common NutionSy is mverthelefi grounded upon injallible Demcnffration. When the Play of the Royal Oak was in ufe, fome Perfons who lojl confiderably by it, had their Lcjjes chiefly occafioned by an Argument of nifnch they could 7iot perceive the Fallacy. The Odds againfi any par- ticular Point of the Ball were One and Thirty to One, which intitled the Adventurers, in cafe they were winners, to have thirty two Stakes returned, including their own ; injlead of which they having but Eight and Twenty, it was very plain that on the /ingle account of the difad- vantage of the Play, they loji one eighth part of all the Money they ' layed for. But the Mafler of the Ball maintained that they had no reafon to complain; fince he woidd undertake that any particular point of the Ball f}.^ould come up in Two and Twenty Throws -, of this he would offer to lay a Wager, and a6lually laid it when required. The feeming contradiBion between the Odds of One and Thirty to One, and Twenty-two ThrowsJor any Chance to come up, fo perplexed the Adventurers, that they begun to think the Advantage was on their fide ; for which reafon they played on and continued to Ife. I The Doftrine of Chances may likewife be a help to cure a Kind of Superftition, which has been oflong ftanding in the World, viz. that there is in Play fuch a thing as Luck, good or bad. I own there are a great many judicious people, who without any other Ajfiftance than that of their 07vn reafon, arefatisfied, that the Notion ofLnck is meerly Chimerical; yet I conceive that the ground they have to look upon it as fuch, may fiill befarther inforcedfrom fome of the following Confiderations.
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