Ex Libris 
 C. K. OGDEN

 
 O F T H E 
 
 I, AW S of C H A N C E, 
 
 M E T* H O D 
 
 Of Calculation of the 
 
 Hazards of GAME, 
 
 Plainly demonftrated, 
 
 And applied to GAMES at prefent 
 inoft in Ufe ; 
 
 Which may be eafily extended to the mofl 
 intricate Cafes of CHANCE imaginable. 
 
 The FOURTH EDITION, 
 Revis'd by JOHN HAM. 
 
 By whom is added, 
 
 A Demonftration of the Gain of the Banker in any 
 Circumftance of the Game call'd PH ARAON ; 
 
 And how to determine the Odds at the 
 ACE of HEARTS or FAIR CHANCE ; 
 
 With the Arithmetical Solution of fome 
 Queftions relating to LOTTERIES; 
 
 And a few Remarks upon HAZARD and 
 BACKGAMMON. 
 
 LONDON: 
 
 Printed for B. MOTTE and C. BAT HURST, at the 
 Middle- temple Gate in Fket-Jireet. M.DC c .xxx v 1 1 1 .
 
 
 
 ' 

 
 [ail 
 
 PREFACE. 
 
 T is thought as neceffary to write 
 a Preface before a Book, as it 
 is judgd civ//, when you invite 
 a Friend to Dinner, to proffer 
 him a Glafs of Hock beforehand for a 
 Whet : And this being maim'd enough 
 for want of a 'Dedication, I am refolv 'd 
 it fball not want an Epiflle to the 
 Reader too. I fhall not take upon me to 
 determine, whether it is lawful to play 
 at *Dice or not, leaving that to be dtf- 
 puted betwixt the Fanatick Parfons and 
 the Sharpers ; / am fure it is lawful to 
 deal with 'Dice as with other Epidemic 
 *Diflemprs ; and I am confident that the 
 writing a Book about it, will contribute ai 
 little towards its Encouragement, 'as Flux- 
 ing and 'Precipitates do to Whoring. 
 
 A 3, It
 
 iv PR EF ACE. 
 
 It will be to little purpofe to tell my 
 Reader, of how great Antiquity the play- 
 ing at T)tce is. I will only let him know-, 
 that by the Alcae Lucius, the Antients com- 
 prehended all Games, which were fubjec- 
 ted to the determination of mere Chance ; 
 this fort of Gaming was fir iff ly forbid by 
 the Emperor Juftinian, Cod. Lib. 3. Tit. 
 43. under very fey ere Penalties', and Pho- 
 cius Nomocan. Tit. 9. Cap. 27. acquaints 
 us, that the Ufe of this was altogether 
 denied the Clergy of that time. Seneca 
 fays very well, Aleator quanto in arte eft 
 melior, tamo eft nequior ; That by how 
 much the one is more skilful in Games, by 
 fo much he is the more culpable; or we 
 may fay of this, as an ingenious Man fays 
 of T)ancingi That to be extraordinary good 
 at it, is to be excellant in a Fault ; there- 
 fore I hope no body will imagine I had fo 
 mean a ^ejign in this, as to teach the Art 
 of T laying at 
 
 A great part of this ^Difconrfe is a 
 Tranflation from Monf. Huygen'j Treatife, 
 De ratiociniis in ludo Alcse ; one, who in 
 his Improvements of 'Philofophy, has but 
 one Superior, and I think few or no K- 
 quals. The whole I undertook for my own 
 tDivertifement, next to the Satisfaction of 
 
 foms
 
 P R E F A C.^E. V 
 
 fome Friends, who would now and then 
 be wrangling about the ^Proportions of 
 Hazards in fome Cafes that are here de- 
 cided. All it requir'd was a few fpare 
 Hours, and but little Work for the Brain ; 
 my T>efign in publishing it, was to make 
 it of more general Ufe, and perhaps per- 
 fuade a raw Squire, by it, to keep his Mo- 
 ney in his ^Pocket -, and if, upon this ac- 
 count, I fhould incur the Clamours of the 
 Sharpers, I do not much regard it, Jince 
 they are a fort of ^People the World is not 
 bound to provide for. 
 
 Ton willfnd here a very plain and eafy 
 Method of the Calculation of the Hazards 
 of Game, which a man may under ft and,with~ 
 out knowing the Quadratures of Curves, 
 the "Doctrine of Series's, or the Laws 
 of Conccntripetation of Bodies, or the 
 ^Periods of the Satellites of Jupiter ; yea, 
 without fa much as the Elements ^Eu- 
 clid. There is nothing required for the 
 comprehending the whole, but common Senfe 
 ana practical Arithmetick ; facing a few 
 Touches of Algebra, as in the frft Three 
 'Proportions, where the Reader, without 
 fufpicion of *Popery^ may make ufe of a 
 ftrong implicit Faith ; tho I muft confefs^ 
 it docs not much recommend it fclf to me 
 in thefe Turpofes ; for / had rather he 
 
 would
 
 vi PREFACE. 
 
 would enquire, and I believe he will frid 
 the Speculation not tinpleafant. 
 
 Every mans Succefs m any Affair is 
 proportional to his Conduct and Fortune. 
 Fortune (in the fenfe of moft f People}Jig- 
 wfies an Event which depends on Chance^ 
 agreeing with my Wi$\ and Misfor- 
 tune fignifies fuch an one, whofe imme- 
 diate Caufes 1 don't know, and consequently 
 can neither for et el nor produce it (for it is 
 no Here/} to believe^ that Providence Buf- 
 fers ordinary matters to run in the Chan- 
 nel of fecond Caufes}. Now I fuppofe, 
 that all a wife Man can do in fuch a Cafe 
 ts y to lay his Buftnefs on fuch Events, as 
 have the moft powerful fecond Caufes, and 
 this is true both in the great Events of 
 the World, and in ordinary Games. It 
 is impojfible for a <D/>, with fuch deter- 
 mirid force and direction, not to fall on 
 fuch a determined fide, only 1 don't know 
 the force and direction which makes it 
 fall on fuch a detennirid Jide, and there- 
 fore I call that Chance, which is no- 
 thing but want of Art -, that only which 
 is left to me, is to wager where there 
 are the greateft number of Chances^ and 
 confidently the greateft probability to 
 gain -, and the whole Art of Gaming, 
 where there is any thing of Hazard, will 
 
 be
 
 PREFACE. vii 
 
 fie r edited to this at lafl, viz, in dubious 
 Cafes to calculate on which fide there are 
 moft Chances j and tho' this can't bt done 
 in the mid ft of Game precifely to an Unitj 
 yet a Man who knows the 'Principles ', may 
 make fitch a conjecture, as will be a fuf- 
 fcient direction to him ; and tho' it is pof- 
 fible, if there are any Chances againft him 
 at all, that he may loft, yet when he chufith 
 the fafeft fide> he may fart with his Mo- 
 ney with more content (if there can be any 
 at all] in fitch a Cafe. 
 
 . I will not debate, whether one may en- 
 gge another in a disadvantageous Wager. 
 Games may be fupposd to be a try at of 
 Wit as well as Fortune, and every Man, 
 when he enttrs the Lifts with another^ 
 unlefs out of Complaifance, takes it for 
 granted^ his Fortune and Judgment, are, 
 at Icaft, equal to thofe of his *P lay -Fellow; 
 but this I am fure of, that falfe Dice, 
 Tricks of Lcger-dc-main, &c. are inex- 
 cufable, for the queftion in Gaming is not y 
 Who is the heft Jnglcr ? 
 
 The Reader may here obferve the Force 
 of Numbers, which can be fuccefsfully ap- 
 plied, even to thofe things, which one 
 would imagine are fubjett to no Rules. 
 There are very few things which we know y 
 
 which
 
 viii PREFACE. 
 
 which are not capable of being reduced to a 
 Mathematical Reafonmg-, and when they 
 cannot , it's a fign our Knowledge of them 
 is very fmall and confus'd 5 and where a 
 mathematical reasoning can be had, it's as 
 great folly to make life of any other, as to 
 grope for a thing in the dark y when you 
 have a Candle flanding by you. I Relieve 
 the Calculation of the Quantity of 'Pro- 
 bability might be improved to a very ufe- 
 ful and pie af ant Speculation, and applied 
 to a great many Events which are acci- 
 dental, befides thofe of Games ; only thefe 
 Cafes would be infinitely more confused, as 
 depending on Chances which the mo ft part 
 of Men are ignorant of-, and as 1 have 
 hinted already, all the Politicks in the 
 JVorld, are nothing elfe but a kind of Ana- 
 lyfis of the Quantity of probability in 
 cafual Events, and a good 'Politician (ig- 
 Tiifies no more? but one who is dextrous at 
 fuchCalculations ; only theTrinciples which 
 are madeufe of in the Solution of Juch J'ro- 
 blems^ cant be ftudied in a Clofet, but ac- 
 quir'd by the Obfcrvatien of Mankind. 
 
 There is likewife a Calculation of the 
 Quantity of 'Probability founded on Ex- 
 perience, to be made ttfe of in JVagers a- 
 bout any thing ; it is odds> if a Woman is 
 
 with
 
 PREFACE. be 
 
 with Child, but it fhall be a Boy 5 and if 
 you would know the juft odds, you muft 
 confider the 'Proportion in the Bills that 
 the Males bear to the Females : The Tearly 
 Bills of Mortality are obfer c u t d to bear fuch 
 ^Proportion to the live People as i to 30, 
 or 26 ; therefore it is an even Wager y 
 that one out of thirteen, dies within a 
 Tear (which may be a good reafon, tho 
 not the true, of that foolifi piece of Super- 
 Jlitiori], becaufe, at this rate, if i out of 
 2.6 dies, you are no lofer. It is but i to 
 18 if you meet a Parfon in the Street , 
 that he proves to be a Non-Juror, becaufe 
 there is but i of $6 that are fuch. It is 
 hardly i to 10, that a Woman of Twenty 
 Tears old has her Maidenhead, and almoft 
 the fame Wager, that a Town -Spark of 
 that Age has not been clap'd. / think a 
 Man might venture fome odds, that 100 
 of the Gens d'arms beats an equal Number 
 of Dutch Troopers; and that an Englifh 
 Regiment ftands its ground as long as ano- 
 ther, making Experience our Guide in all 
 thefe Cafes and others of the like nature. 
 
 But there are no cafual Events, which 
 are fo eafily fubjeffed to Numbers, as thoft 
 of Games ; and I believe, there the Specu- 
 lation might be improved fo far, as to bring 
 in the Doctrine of the Series'.? and Loga- 
 a rithms.
 
 x PREFACE. 
 
 richms. Since Gaming is become a Trade, 
 1 think it ft the Adventurers fiould be 
 upon the Square ; and therefore in the Con- 
 trivance of Games there ought to be aftrift 
 Calculation made ufe of] that they mayn't 
 put one Tarty in more probability to gain 
 than another ; and likewife, if a Man has 
 a conjtderable Venture ; he ought to be al- 
 low d to withdraw his Money when he 
 pleafes, paying according to the Circum- 
 Jtances he is then in : and it were eafy in 
 moft Games to make Tables, by Infpeffion 
 of which, a Man might know what he 
 was either to pay or receive, in any Cir- 
 cumjlances you can imagin-, it being conve* 
 Silent to fave apart of ones Money, rather 
 than venture the lofs of it all. 
 
 I fa-all add no more, but that a Ma- 
 . thematician will eajily perceive, it is not 
 .put in fuc h a Drefs as to be taken notice 
 
 oj by him, there being abundance of Words 
 fpent to make the more ordinary fort of 
 
 'People under jland it.
 
 FOR the fake of thofe who are not 
 vers'd in Mathematicks, I have added 
 the following Explanation of Signs. 
 
 = Equal. 
 
 4- More, or to be added. 
 
 Lefs, or to be fubtra&ed. 
 
 x Multiplied. 
 
 -f- Divided. 
 
 EXAMPLE.' 
 
 3 x4-f- 3 i = i4 = |tf, is to be 
 read thus $ 
 
 3 multiplied in 4, more by 3, lefs by i] 
 is equal to 14, which is equal to five 
 ninth parts of a.
 
 An EXACT 
 
 METHOD 
 
 For S O LV I N G the 
 
 HAZARDS of G A M E. 
 
 LTHO' the Events of Games, 
 which Fortune folely governs, are 
 uncertain, yet it may be certainly 
 determin'd, how much one is 
 more ready to lofe than gain. "For Example: 
 If one mould wager, at the firft throw with 
 one Die, to throw fix, it's an accident if he 
 gains or not ; but by how much it's more 
 probable he will lofq than gain, is really de~ 
 termin'd by the Nature of the thing, and 
 capable of a ftricl: Calculation. So likewife 
 if I mould play with another on tfyis con- 
 dition, that the Victory mould be to the 
 three nrft Games, and I had gain'd one al- 
 B ready,
 
 2 Solution of the 
 
 ready, it is flill uncertain who mall gain 
 the third j yet by a demonftrative Reafon- 
 ing, I can eflimate both the Value of his 
 Expectation and mine, and confequently (if 
 we agree to leave the Game imperfect) deter- 
 mine how great a mare of the Stakes belong 
 to me, and how much to my Play-fellow ; 
 or if any were defirous to take my place, 
 at what rate I ought to fell it. Hence may 
 arife innumerable Queries among two, three, 
 or more Gameflers : and lince the Calcula- 
 tion of thefe things is a little out of the 
 common Road, and can be oft-times apply'd 
 to good purpofe, I fhall briefly here ihew 
 how it is to be done, and afterwards ex- 
 plain thofe things which belong properly to 
 the Dice. 
 
 In both Cafes I fhall make ufe of this 
 Principle, Ones Hazard or Expectation to 
 gain any thing, is worth fo much, as, if he 
 had it, he could pur chafe the like Hazard or 
 Expectation again in a jujl and equal Game. 
 
 For Example, if one, without my Know- 
 ledge, mould hide in one hand 7 Shillings, 
 and in his other 3 Shillings, and put it to 
 my choice which Hand I would take, I fay 
 this is as much worth to me, as if he mould 
 give me 5 Shillings ; becaufe, if I have 5 
 Shillings, I can purchafe as good a Chance 
 again, and that in a fair and juft Game. 
 
 PRO-
 
 HAZARDS of Game. 3 
 
 PROPOSITION I. 
 
 If I expett a or b, either of which, with 
 equal probability ', may fall to me, then my 
 
 Expectation is worth *t- y that is, the 
 half Sum of a and b. 
 
 THAT I may not only demonflrate, 
 but likewife invefligate this Rule, 
 fuppofe the Value of my Expectation be x-, 
 by the former Principle having x, I can 
 purchafe as good an Expectation again in a 
 fair and juft Game. Suppofe then I play 
 with another on thefe terms, That every 
 one flakes x, and the Gainer give to the 
 Lofer a, this Game is juft ? and it appears, 
 that at this rate, I have an equal hazard 
 either to get a if I lofe the Game, or 2x^-a 
 if I gain ; for in this cafe I get 2X, which 
 are the Stakes, out of which I muft pay 
 the other a ; but if 2x a were worth k 9 
 then I have an equal hazard to get a or b ; 
 
 therefore making 2x ==^, #== 7-' 
 which is the Value of my Expectation. 
 The Demonstration is eafy ; for having 
 
 a ^ , I can play with another who will 
 
 flake ^~- againfl it, on this condition, 
 
 that the Gainer fhould give to the Lofer a; 
 B 2 by
 
 4 Solution of tie 
 
 by this means I have an equal Expectation 
 to get a if I lofe, or b if I win ; for in the 
 laft cafe I get d-\-b the Stakes, out of 
 which I muft pay a to my Play-fellow. 
 
 In Numbers : if I had an equal hazard 
 to get 3 or 7, then by this Proportion, 
 my Expectation is worth 5, and it is cer- 
 tain, having 5, 1 may have the fame Chance ; 
 for if I play with another, fo that every 
 one flakes 5, and the Gainer pay to the 
 Lofer 3, this is a fair way of gaming j and 
 it is evident I have an equal hazard to get 
 3 if I lofe, or 7 if I gain. 
 
 PROP. II. 
 
 IJ I expett a, b, ore, either of which, 'with 
 equal facility ^ may happen ^ then the Value 
 
 of my Expectation is - H , or the, 
 
 third part of the Sum of& 3 b, and c. 
 
 FO R the Inveftigation of which, fup- 
 pofe x be the value of my Expecta- 
 tion j then x muft be fuch, as I can pur- 
 chafe with it the fame Expectation in a 
 jufl Game: Suppofe the Conditions of the 
 Game be, that playing with two others, 
 each of us flakes x, and I bargain with one 
 of the Gameflers, if I win, to give him b, 
 andrhe fhall do the fame to me 5 but with 
 
 the.
 
 HAZARDS of Game. 5 
 
 the other, that if I gain, I (hall give him c > 
 and vice versa ; this is fair play : And here 
 I have an equal hazard to get b y if the firft 
 win, c if the fecond, or 3 AT c if I 
 gain myfelf ; for then I get 3*, viz. the 
 Stakes, of which I give the one b and the 
 other c ; but if 3* b c be equal to 
 a, I have an equal Expectation of a, b y 
 or c therefore making 3* b c = a, 
 
 t which is the Value of my 
 
 After the fame Method you 
 will find, if I had an equal hazard to get 
 a, b y c, or d, the Value of my Expectation 
 + +c + tl y that is the fourth part of the 
 Sum of a, b t c, and d, &c. 
 
 PROP.
 
 6 Solution of the 
 
 PROP. III. 
 
 y the number of Chances, by which a falls to 
 me, be p, and the number of Chances, by 
 which b falls, be q, and fuppofing all the 
 Chances do happen with equal facility, then 
 
 the Value of my Expectation is lii q 5 i. e . 
 
 the Product of a multiplied in the number 
 cf its Chances added to the Product of b, 
 multiplied into the number of its Chances, 
 end the Sum divided by the number of 
 Chances both of a and b. 
 
 SUppofe, as before, x be the Value of 
 my Expectation ; then if I have x, I 
 muft be able to purchafe with it that fame 
 Expectation again in a fair Game : For 
 this I mall take as many Play-fellows as, 
 with me, make up the number of p -f- q, 
 of which let every one flake x, fo the 
 whole Stake will be px 4- qx, and every 
 one plays with equal hopes of winning ; 
 with as many of my Fellow- Gamefters as 
 the Number q ftands for, I make this bar- 
 gain one by one, that whoever of them 
 gains #iall give me b, and if I win, I mall 
 do fo to them ; with every one of the reft 
 of the Gamefters, whofe Number is/> i, 
 I make this bargain, that whoever of them 
 
 gains,
 
 HAZARDS of Game. 7 
 
 gains, (hall give me a, and I {hall give 
 every one of them as much, if I gain : 
 It's evident this is fair play ; for no Man 
 here is injur'd j and in this cafe I have 
 q Expectations to gain b y and p i Ex- 
 pectations to gain #, and i Expectation 
 (viz. when I win myfelf ) to get px -+- qx 
 
 bq a p -f- a ; for then I am to deliver 
 b to every one of the q Players, and a to 
 every one of the p i Gameflers, which 
 makes qb-\-pa a ; if therefore qx -f- bx 
 
 bq ap -+- a were equal to a, I would 
 have/> Expectations of a (fince juft now I had 
 p i Expectations of it) and ^Expecta- 
 tions of b, and fo would have juft come to 
 my nrft Expectation ; therefore putting 
 px-\-qx bq ap-\-a = a, then is x = 
 
 ap -4- bq 
 f +1 
 
 In Numbers : If I had 3 Chances to gain 
 for 13, and 2 for 8, by this Rule, my ha- 
 zard is worth ii; for 1 3 multiplied by 3 
 gives 39, and 8 by 2 1 6, thefe two added, 
 make 55, divided by 5 is iij and I can 
 eafily ihew, if I have n, I can come to 
 the like Expectation again ; for playing 
 with four others, and every one of us 
 flaking u, with two of them I make this 
 bargain, that whoever gains fhall give me 
 8, and I fhall too do fo to them ; with the 
 other two I make this bargain, that who- 
 ever
 
 8 Solution of the 
 
 ever gains mall give me 13, and I them as 
 much if I gain : it appears, by this means 
 I have two Expectations to get 8, viz. 
 if any of the firft two gain, and 3 Expec- 
 tations to get 13, viz. if either I or any 
 of the other two gain ; for in this cafe I 
 gain the Stakes, which are 55, out of 
 which I am oblig'd to give the firft two 8, 
 and the other two 13, and fo there re- 
 mains 13 for myfelf. 
 
 PROP. IV. 
 
 fflat I may come to the Queftion proposed, 
 viz. 'The making a juft Diftribution a- 
 mongft Gamefters, 'when their Hazards 
 are unequal } ive muft begin with the moft 
 eafy Cafes. 
 
 SUppofe then I play with another, on 
 condition that he who wins the three 
 firit Games mall have the Stakes, and that 
 I have already gain'd two, I would know, 
 if we agree to break off the Game, and 
 part the Stakes juftly, how much falls to 
 my mare ? 
 
 The firft thing we muft confider in fuch 
 Queftions is the number of Games that 
 are wanting to both : For Example, if it 
 had been agreed betwixt us, that he mould 
 have the Stakes who gain'd the firft 20 
 
 Games,
 
 HAZARDS of Game. Cy 
 
 Games, and if I had gain'd already 19, 
 and rny Fellow-Gamefler but 18, my ha- 
 zard is as much better than his in that cafe, 
 as in this propofed, viz. When of 3 Games 
 I have 2, and he but one, becaufe in both 
 cafes there's 2 wanting to him, and i to 
 me. 
 
 In the next place, to find the portion of 
 the Stakes due to each of us, we muft 
 confider what would happen if the Game 
 went on j it is certain, if I gain the firft 
 Game, I get the Stake, which I call a , 
 but if he gain'd, both our Lots would be 
 equal, and fo there would fall to each of 
 us ~a -, but iince I have an equal hazard 
 to gain or lofe the firfl Game, I have an 
 equal Expectation to gain a, or a, which, 
 by the firft Propofition y is as much worth as 
 the half Sum of both, /. e. %a, fo there is 
 left to my Fellow-Gamefter a ; from 
 whence it follows, that he who would buy 
 my Game, ought to pay me for it %a > and 
 therefore, he who undertakes to gain one 
 Game before another gains two, may wager 
 3 to I ' 
 
 PROP;
 
 liO Solution of the 
 
 PHOP. V. 
 
 Suppofe Iivant but one Game, and my Fellow-* 
 Gamefier three, it is required to make a 
 juji ibijlribution of the Stake. 
 
 LE T us here likewife confider in what 
 ftate we mould be, if I or he gain'd 
 the firft Game j if I gain, I have the Stake 
 a, if he, then he wants yet 2 Games, and 
 I but i, and therefore we mould be in the 
 fame Condition which is fuppofed in the 
 former Proportion j and fo there would fall 
 to my Share, as was demonftrated there, 
 ^a ; therefore with equal facility there may 
 happen to me a, or A^ which, by the firft 
 Proportion, is worth ^, and to my Fel- 
 low-Gamefter there is left -#, and there- 
 fore my hazard to his is as 7 to i. 
 
 As the Calculation of the former Propo- 
 fition was requifite for this, fo this will 
 ferve for the following. If I mould fup- 
 pofe myfelf to want but one Game, and 
 my Fellow four, (by the fame Method) 
 you will find -j-f of the Stake belongs to- 
 me, and -^ to him. 
 
 PROP.
 
 HAZARDS of Game. \ t 
 
 PROP. VI. 
 
 Suppofe I want two Games, and my Fellow* 
 Gamejler three. 
 
 THEN by the next Game it will hap- 
 pen that I want but one, and he 
 three, which (by the preceding Proportion) 
 is worth ^a j or that we mould both want 
 two, whence there will be \a due to each 
 of us : now I being in an equal probability 
 to gain or lofe the next Game, I have an 
 equal hazard to 'gain ^a or \a, which by 
 the firft Proportion is worth ^a j and fo 
 there are eleven parts of the Stakes due to 
 me, and five to my Fellow. 
 
 PROP. VII. 
 
 Let us fuppoje I want two Games, and my 
 Fellow four. 
 
 IF I gain the next Game, then I mall 
 want but one, and my Fellow four -, 
 but if I lofe it, then I {hall want two, and 
 he three : So I have an equal hazard for 
 gaining ^-f a t or 4.^7, which, by the firfr, 
 is worth -iftf : So it. appears, that he who is 
 to gain two Games ior the other's four, i 
 in a better condition than he who is to. 
 C 2
 
 t % Solution of tie 
 
 gain one for the other's two ; for my {hare 
 in the firfl cafe is ^a or LLa y which is lefs 
 than ~L ) my mare in the laft. 
 
 PROP. VIII. 
 
 Let us fuppofe three Gamefters, whereof the 
 Jirji and fecond 'want i Game, but the 
 third 2. 
 
 TO find the mare of the fir ft, we muft 
 confider what would happen if ei- 
 ther he, or any of the other two gain'd the 
 firft Game j if he gains, then he has the 
 Stake a ; if the fecond gain, he has nothing; 
 but if the third gain, then each of them 
 would want a Game, and fo La would be 
 due to every one of them. Thus the firft 
 Gamefter has one Expectation to gain a y 
 one to gain nothing, and one for La, (fince 
 all are in equal probability to gain the firft 
 Game) which by the fecond Propofition is 
 worth a : Now fince the fecond Gamefter's 
 Condition is as good, his Share is like wife 
 %a, and fo there remains to the third ^a, 
 whofe Share might have been as eafily found 
 by itfelf.
 
 HAZARDS of Game, 13 
 
 PROP. IX. 
 
 In any number of Gamefters you pleafe, a- 
 mongft whom there are fome who want, 
 more, fome fewer Games : To find what 
 is any one's Jhare in the Stake, we muft 
 conjider what would be due to him, whofe 
 Share we inveftigate, if either he, or any 
 of his Fellow-GameJiers fiould gam the 
 next following Game j add all their Shares 
 together, and divide the Sum by the num- 
 ber of the Gamejlers, the Quotient is his 
 Share you were feeking. 
 
 SUppofe three Gamefters A, B, and C ; 
 A wants i Game, B 2, and Clikewife 
 2, I would find what is the fliare of the 
 Stake due to B, which I {hall call q. 
 
 Firft, we muft confider what would fall 
 to B's Share, if either he, A, or C, wins 
 the next Game ; if A wins, the Game is 
 ended, fo he gets nothing ; if B himfelf 
 gain, then he wants i Game, A i, and 
 C 2 ; therefore, by the former Proportion, 
 there is due to him in that cafe %q, then 
 if C gains the next play, then A and C 
 would want but i, and -B 2; and there- 
 rfore, by the eighth Propofition, his Share 
 would be worth ^q ; add together what is 
 due to B in all thefe three Cafes, viz. o|f ,
 
 14- Solution of the 
 
 ^.q, the Sum is |^, which being divided 
 by 3, the number of Gamefters, gives ^q, 
 which is the Share of B fought for : The 
 Demonftration of this is clear from the 
 fecond Propofition, becaufe B has an equal 
 hazard to gain c$q or L^ that is "*rfo"H y > 
 
 i. e. -f\q : now it's evident the Divifor 3 is 
 the number of the Gamefters. 
 
 To find what is due to one in any cafe j 
 viz. if either he, or any of his Fellow- 
 Gamefters win the following Game ; we 
 rnuft confider firft the more fimple Cafes, 
 and by their help the following j for as 
 this Cafe could not be folv'd before the 
 Cafe of the eighth Propofition was calcu- 
 lated, in which, the Games wanting were 
 i, i, 2 ; fo the Cafe, where the Games 
 wanting are i, 2, 3, cannot be calculated, 
 without the Calculation of the Cafe, where 
 the Games wanting are i, 2, 2, (which we 
 have juft now perform'd) and likewife of 
 the Cafe, where the Games wanting are 
 i, i, 3, which can be done by the eighth: 
 And by this means you may reckon all the 
 Cafes comprehended in the following Tables, 
 and an infinite number of others. 
 
 Games
 
 HAZARDS of Game. 
 
 Games wanting 
 Their Shares 
 
 Games wanting 
 Shares 
 
 i,l,Z\ 1,2,2, 
 
 4,4,1 
 
 ',3, 
 
 3,n.i.liQ,6,3, 
 
 *7 
 
 a 7 
 
 , 40,1 1 121, 1 21,1 li 78,158,7 1 j;42, 179, 
 
 243 
 
 729 
 
 Games wanting 1,3,3} i, 3, 4 
 
 Their Shares |6 ; , 8, 81616,82,31 
 
 629,87,13 
 
 729 
 
 729 
 
 Games wanting 
 Their Shares 
 
 Games wanting 
 Their Shares 
 
 729 
 
 729 
 
 2. 1, 3 
 
 2, 3, 4 
 
 2, 3 f 
 
 '33, 55,^5 
 
 4>,i95> 8 3 
 
 433,635, 1 19 
 
 243 
 
 729 
 
 1187 
 
 As for the Dice; thefe Queftions may 
 be propofed, at how many Throws one 
 may wager to throw 6, or any Number 
 below that, with one Die j How many 
 Throws are required for 1 2 upon two Dice ; 
 or 1 8 on 3 j and feveral other Queftions to 
 this purpofe. 
 
 For the refolving of which, it muft be 
 confider'd, that in one Die there are fix 
 different Throws, all equally probable to 
 come up ; for I fuppofe the Die has the 
 
 exaft
 
 1 6 Solution of tie 
 
 exact figure of a Cube : On two Dice 
 there are 36 different throws ; for in re^ 
 fpedt to every throw of one Die, any one 
 throw of the 6 of the other Die may come 
 up; and 6 times 6 make 36. In three 
 Dice there are 216 different throws; for 
 iii relation to any of the 36 throws of 
 two Dice, any one of the fix of the third 
 may come up; and 6 times 36 make 216: 
 So in four Dice there are 6 times 216 
 throws, that is, 1296 : And fo forward 
 you may reckon the throws of any num- 
 ber of Dice, taking always, for the Addi- 
 tion of a new Die, 6 times the number of 
 the preceding. 
 
 Befides, it mufl be obferv'd, that in two 
 Dicfe there is only one way 2 or 12 can 
 come up ; two ways that 3 or 1 1 can come 
 up ; for if I fhall call the Dice A and B, to 
 make 3 there may be i in A, and 2 in B, 
 or 2 in A, and i in B ; fo to make 1 1, 
 there may be 5 in A, and 6 in B, or 6 in A, 
 and 5 in B ; for 4 there are three Chances, 
 3 in A, and i in B, 3 in B, and i in A, 
 or 2 as well in A as B ; for i o there are 
 'Jikewife three Chances ; for 5 or 9 there 
 are four Chances ; for 6 or 8 five Chances ; 
 for 7 there are fix Chances. 
 
 I In
 
 HAZARDS of Game. 17 
 
 ' 3 or 18 
 
 i 
 
 4 or 17 
 5 or 1 6 
 6 or 15 
 
 3 
 6 
 
 10 
 
 7 or 14 
 8 or 13 
 
 '5 
 
 21 
 
 9 or 12 
 
 2 5 
 
 . 10 or ii 
 
 2 7 
 
 In 3 Dice there are found 
 for 
 
 PROP. X. 
 
 To find at how many times one may under-^ 
 take to throw 6 with one Die. 
 
 IF any mould undertake to throw 6 the 
 firft time, it's eviden Where's one Chance 
 gives him the Stake, and five which give 
 him nothing; for there are 5 throws a- 
 gainft him, and only one for him. Let 
 the Stake be call'd a, then he has one Ex- 
 pectation to gain #, and five to gain no- 
 thing, which, by the third Proportion, is 
 worth 0, and there remains for the other 
 tf j fo he who undertakes, with one Dif, 
 to throw 6 the firft time, ought to wager 
 only i to 5. 
 
 2. Suppofe one undertake, at two Throws 
 of i Die, to throw 6, his Hazard is cal- 
 culated thus ; if he throw 6 at the firft, 
 he has a the Stake ; if he do not, there re- 
 
 mains
 
 1 8 Solution of the 
 
 mains to him one throw, which, by the 
 former Cafe, is worth ^a ; but there is but 
 one Chance which gives him 6 at the firft 
 throw, and five Chances againft him; fo 
 there is one Chance which gives him a> 
 and five which give him .*, which by the 
 fecond Pro^ofition, is worth J^a, fo there 
 remains to his Fellow-Gamefter * -, fo the 
 Value df my Expectation to his, is as n to 
 25, /. . lefs than i to 2. 
 
 By the fame method of Calculation, you 
 will find, that his hazard who undertakes 
 to throw 6 at three times with one Die, 
 is T 9 T V tf *~ tnat he can only lay 9 1 a- 
 gainft 125, which is fomething lefs than 3 
 to 4. 
 
 He who undertakes to do it at four times, 
 his hazard is T VyV^ f he may wager 67 r 
 againft 625, that is, fomething more than 
 
 1 to i. 
 
 He who undertakes to do it at five times, 
 his hazard is yff j-0, fo he can wager 4651 
 againft 3125, that is, fomething lefs than 
 
 3 to . 2 - 
 
 His hazard who undertakes to do it at 6 
 
 times, is ^-f-r tf > an ^ ^ e can wa g er 3 IO 3 r 
 againft 15625, that is, fomething leis than 
 
 2 to i. 
 
 Thus any number of throws may be eafily 
 found ; but the following Propoiition will 
 fhew you a more compendious way of Cal- 
 culation, PROP.
 
 HAZARDS of Gam& 1 9 
 
 PROP. XT. 
 
 To Jlnd at hoiu many times one may undertake 
 to thrt/iv 1 2 with two Dice. 
 
 IF one fhould undertake it at one throw, 
 it's clear he has but one Chance to get 
 the Stake a, and 35 to get nothing ; which, 
 by the third Proportion, is worth ~ r a. 
 
 He who undertakes to do it at twice, if he 
 throw 1 2 the firft time, gains a j if otherwife, 
 then there remains to him one throw, 
 which, by the former Cafe, is worth ~^a ; 
 but there is but one Chance which gives 12 
 at the firft throw, and 35 Chances again ft 
 him j fo he has i Chance for a, and 35 for 
 Ts- a > which by the third Proportion is 
 worth T-J-JT^J and there remains to his 
 Fellow-Gamefter 44^. 
 
 From thefe it's eafy to find the Value of 
 his hazard, who undertakes it at four times, 
 palling by his cafe who undertakes it at 
 three times. 
 
 If he who undertakes to do it at four 
 times throws 12 the firfl or fecond Caft, 
 then he has a -, if not, there remains two 
 other throws, which, by the former Cafe, 
 are worth T ^ T a $ but for the fame reafon, 
 in his two firit throws, he has 7 1 Chances 
 which give him a, againft 1225 Chances, 
 D 2 i*
 
 20 Solution of the 
 
 in which it may happen otherwife j there- 
 fore at firft he has 71 Chances which give 
 him a, and 1225 which give him -r^-J-ytf, 
 which by the third Proportion is worth, 
 ^JLy.. 1 ^ which mews that their hazards 
 to one another are as 178991 to 1500625. 
 
 From which Cafes it is eafy to find the 
 Value of his Expe&ation, who undertakes 
 to do it at 8 times, and from that, his Cafe 
 who undertakes to do it at 1 6 times ; and 
 from his Cafe who undertakes to do it at 8 
 times, and his likewife who undertakes to 
 do it at 1 6 times ; it is eafy to determine 
 his Expectation who undertakes it at 24 
 times: In which Operation, becaufe that 
 which is principally fought, is the number 
 of throws, which makes the hazard 'equal 
 on both fides, viz. to him jyvho under- 
 takes, and he who offers, you may with- 
 out any fenfible Error, from the Numbers 
 (which elfe would grow very great) cut off 
 fome of the laft Figures. And fo I find, 
 that he who undertakes to throw 12 with 
 two Dice, at 24 times, has fome lofs ; and 
 he who undertakes it at 25 times, has fome 
 advantage. 
 
 PROP*
 
 HAZARDS of Game. 2t 
 
 PROP. XII. 
 
 tfojind with how many Dice one can under*, 
 take to throw two Sixes at the firft Caft. 
 
 THIS is as much, as if one would 
 know, at how many throws of one 
 Die, he may undertake to throw twice 
 fix : now if any fhould undertake it, at 
 two throws, by what we have fhewn be- 
 fore,, his hazard would be T V*j he who 
 would undertake to do it at 3 times, if his 
 firft throw were not 6, then there would 
 remain two throws, each of which muft 
 be 6, which (as we have {aid) is worth ^a ; 
 but if the firft throw be 6, he wants only 
 one 6 in the two following throws, which 
 by the tenth Propofition, is worth 0-^a : 
 but fince he has but one Chance to get 6 
 the firft throw, and five to mifs it ; he has 
 therefore, at firft, one Chance for ^a, and 
 five Chances-for - l ^a, which, by the third 
 Propofition, is worth T VV*> or -sr a > a ^ ter 
 this manner ftill afluming i Chance more, 
 you will find that you may undertake to 
 throw two Sixes at 10 throws of one Die, 
 or i throw of ten Dice, and that with fome 
 advantage. 
 
 PROP.
 
 Solution of the 
 
 PROP. XIII. 
 
 Jflam to play with another one Throw, on 
 this condition, that if j comes up I gain, 
 if 10 he gains ; if it happens that we muft 
 divide the Stake, and not play, to find 
 how much belongs to me, and how much 
 to him. 
 
 BEcaufe of the 36 different Throws of 
 the two Dice, there are fix which 
 give 7, and 3 which give 10, and 27 which 
 equals the Game, in which cafe there is 
 due to each of us \a : But if none of the 
 27 fhould happen, I have 6, by which I 
 may gain a, and 3, by which I may get 
 nothing, which by the third Proportion, 
 is worth ^a ; fo I have 27 Chances for a, 
 and 9 for ^a, which, by the third Propo- 
 iition, is worth ~^a, and there remains to 
 my Fellow-Gameiler ~^a. 
 
 PROP,
 
 HAZARDS of Game. 23 
 
 PROP. XIV. 
 
 If I were playing with another by turns, with 
 two Dice, on this condition^ that if I throw 
 7 I gain, and if he throw 6 be gains, al- 
 lowing him the firjl throw : To find the pro- 
 portion of my Hazard to his. 
 
 SUppofe I call the Value of my Hazard 
 x, and the Stakes a, then his Hazard 
 will be a x ; then whenever it's his turn 
 to throw, my Hazard is x, but when it*s 
 mine, the Value of my Hazard is greater. 
 Suppofe I then call it y ; now becaufe of 
 the 36 throws of two Dice, there are five 
 which give my Fellow-Gamefter 6, thirty- 
 one which bring it again to my turn to 
 throw, I have five Chances for nothing, 
 and thirty-one for y, which, by the third 
 Propofition, is worth A^-y j but I fuppos'd 
 at firft my Hazard to be x ; therefore 
 |y s= x, and confequently y = .|4*- I 
 fuppos'd likewife, when it was my turn 
 to throw, the Value of my Hazard was y ; 
 but then I have fix Chances which give 
 me 7, and confequently the Stake, and 
 thirty which give my Fellow the Dice, 
 that is, make my Hazard worth x ; fo I 
 have fix Chances for a, and thirty for x, 
 
 which,
 
 24 Solution of the 
 
 which, by Propofit. 3. is worth 
 
 but this by fuppofition is equal to y, which 
 is equal (by what has been prov'd already) 
 to-g*, therefore 2^ == **, and 
 
 confequently x = ^^a y the Value of my 
 Hazard, and that of my Fellow-Gamefter 
 is J4#, fo that mine is to his as 3 1 to 30. 
 
 Here follow feme ^uejiions which ferve to 
 exercije the former Rules. 
 
 1. A and B play together with two Dice, 
 A wins if he throws 6, and B if he throws 
 7 ; A at firft gets one throw, then B two, 
 then A two, and fo on by turns, till one 
 of them wins. I require the proportion of 
 A s Hazard to B's ? Anfwer, It is as 1 03 55 
 to 12276. 
 
 2. Three Gamefters, A, B, and C, take 
 12 Counters, of which there are four white 
 and eight black ; the Law of the Game is 
 this, that he fhall win, who, hood-wink'd, 
 (hall firft chufe a white Counter ; and that 
 A fhall have the firft choice, B the fecond, 
 and C the third, and fo, by turns, till one 
 of them win. Quar. What is the propor- 
 tion of their Hazards ?
 
 HAZARDS of Game. 2$ 
 
 3. ^wagers with B, that of 40 Cards, 
 that is, 10 of every Suit, he will pick out 
 four, fo that there (hall be one of every 
 fuit. A's Hazard to B's in this cafe is as 
 1000 to 8139. 
 
 4. Suppofing, as before, 4 white Coun- 
 ters and 8 black, A wagers with B, that 
 out of them he (hall pick 7 Counters, of 
 which there are. 3 white. I require the 
 proportion of A's Hazard to J5's ? 
 
 5. A and B taking 12 Counters, each 
 play with three Dice after this manner, that 
 if 1 1 comes up, A (hall give one Counter 
 to By but if 14 comes up, B mail give one 
 to A^ and that he lhall gain who firft has 
 all the Counters. A's Hazard to jB's is 
 244140625 to 282429536481. 
 
 The Calculus of the preceding Problems 
 is left out by Monf. Huygens, on purpofe 
 that the ingenious Reader may have the 
 fatisfa&ion of applying the former Method 
 himfelf ; it is in moft of them more labo- 
 rious than difficult : for Example, I have 
 pitch'd upon the fecond and third, be- 
 caufe the reft can be folv'd after the fame 
 Method. 
 
 6 PROBLEM
 
 26 Sdut'wn of tb& 
 
 PROBLEM i. 
 
 The firft Problem is folv'd by the Me- 
 thod of Prop. 14. only with this difference, 
 that after you have found the Share due to 
 B 3 if A were to get no firft throw, you 
 muft fubtracl from it T s r of the Stake which 
 is due to A for his Hazard of throwing fix 
 at the Jfirft throw. 
 
 PROBLEM 2. 
 
 As for the fecond Problem, it is folv'd 
 thus 5 Suppofe jfs Hazard, when it is his 
 own turn to chufe, be x y when it is 5's, be 
 j, and when it is C*s, be z -, it is evident, 
 when out of 1 2 Counters, of which there 
 are 4 white and 8 black, he endeavours to 
 chufe a white one, he has four Chances 
 to get it, and eight to mifs it ; that is, he 
 has four Chances to get the Stake a, and 
 eight to make his hazard worthy : fox =2 
 ^> , and confequently y = 1 2L=^ - 
 
 When it is B's turn to chufe, then he has 
 four Chances for nothing, and eight for 
 'z, (that is, to bring it to C's turn) confe- 
 quently y = ^z = l~p 4 - ; this Equa- 
 reduc'd gives K= Q . x ~^ a . w hen it 
 
 comes
 
 HAZARDS of Game. 27 
 
 comes to (7s turn to chute, then A has four 
 Chances for nothing, and eight for x, con- 
 
 fequently z=^x, therefore ^x 
 
 this equation reduc'd gives x =.^a y and 
 confequently there remains to B and C -l-Jtf, 
 which muft be fhar'd after the fame man- 
 ner, that is, fo that B have the firft choice, 
 C the next, and fo on, till one of them 
 gain $ the reafon is, becaufe it had been 
 juft in A to have demanded T ^. of the 
 Stake for not playing, and then the Se- 
 niority fell to J5 > now ? a > parted be- 
 twixt B and C', by the former Method, 
 gives T ^ to By and T 4 T to C j fo A^ B, and 
 C"s Hazards from the beginning were as 
 
 9* 6 > 4- 
 
 I have fuppos'd here the Senfe of the 
 Problem to- be, that when any one chus'd 
 a Counter, he did not diminim their num- 
 ber ; but if he mifs'd of a white one, put 
 it in again, and left an equal hazard to 
 him who had the following choice ; for if 
 it be otherwife fuppos'd, A's fhare will be 
 T y T) which is lefs than T ^.. 
 
 Prob. 2. It is evident, that wagering to 
 pick out 4 Cards out of 40, fo that there 
 be one of every Suit, is no more than wa- 
 gering, out of 39 Cards to take 3 which 
 fhall be of three propofed Suits ; for it is 
 all one which Card you draw firft, all the 
 E a hazard
 
 28 Solution of the 
 
 hazard being, whether put of the 3 9 re- 
 maining you take 3, of which none fhall be 
 of the Suit you firft drew : Suppofe then 
 you had gone right for three times, and 
 were to draw your laft Card, it is clear 
 that there are 27 Cards, (viz. of the Suits 
 you have drawn before) of which, if you 
 draw any you lofe, and 10, of which if 
 you draw any, you have the Stake a j fo 
 you have 10 Chances for a, and 27 for 
 nothing, which, by Prop. 3. is worth yftf. 
 Suppofe again you had gone right only for 
 two Draughts, then you have 18 Cards 
 (of the Suits you have drawn before) which 
 make you lofe, and 20, which put you 
 in the Cafe fuppos'd formerly, viz. where 
 you have but one Card to draw, which, 
 as we have already calculated, is worth 
 JL^-a $ fo you have 18 Chances for nothing, 
 and 20 for JL*.a, which, by Prop. 3. is 
 worth y|-|tf. "Suppofe again you have 3 
 Cards to draw, then you have 9 (of the 
 Suit you drew firft) which make you lofe, 
 and 30 which put ycu in the cafe fuppos'd 
 laft ; fo you have 9 Chances for nothing, 
 and 30 for f|Jrf, which by Propof. 3. is 
 worth -J T y_? T <j, or ^~? a > an ^ y ou leave to 
 your Fellow-Gamelter |4j|- a ; fo your ha- 
 zard is to his as 1000 to 8139. 
 
 It is eafy to apply this Method to the 
 Games that are in ufe amongft us : For 
 
 Example^
 
 HAZARDS of Game. 29 
 
 , If A and 5, playing at Backgam- 
 mon, A had already gain'd one end of three, 
 and B none, and if A had the Dice in his 
 Hand for the laft throw of the fecond end, 
 all his Men but two upon the Ace Point 
 being already caft off: $u<zr. What is the 
 proportion of As Hazard to B's ? 
 
 Solution: There being of the 36 Throws 
 of two Dice, fix which give Doublets ; if 
 A throw any of the fix, he has the flake a $ 
 if he throw any of the other thirty, then 
 he wants but one Game, and his Fellow- 
 Gamefter three, which, by Prop. V. is \a j 
 fo A has fix Chances for a y and thirty for 
 -J-j, which, by Prop. III. is worth rJi* an d 
 there remains to his Play-Fellow rVV* ; fo 
 A's Hazard to B's is as 129 to 15, that is, 
 lefs than 9 to i. 
 
 Suppofing the fame Cafe, and if thek- 
 Bargain had been, that he who gain'd three 
 ends before the other gain'd one, mould 
 have double of what each ftak'd, that is, 
 the Stake and a half more, then there had 
 been due to A HT f r ^ e Stake, that is, 
 E ought only to take T V> and leave the refl 
 to A. 
 
 Thus likewife, if you apply the former 
 Rule to the Royal-Oak Lottery, you will 
 find, that he who wagers that any Figure 
 fhall come up at the firfl throw, ought to 
 wager i againfl 31 ; that he who wagers 
 
 it
 
 30 Solution of the 
 
 it fhall come up at one of two throws^, 
 ought to wager 63 againft 961 ,- that he 
 who wagers that a Figure (hall come up at 
 once in three times, ought to lay 125055 
 againft 923521, Gfr. it being only fome- 
 what tedious to calculate the reft. Where 
 you will find, that the equality will not 
 fall as fome imagine on 16 Throws, no 
 more than the equality of wagering at how 
 many Throws of one Die 6 mall come up, 
 falls on three 5 the contrary of which you 
 have feen already demonftrated : you will 
 find by calculation, that he has the Dif- 
 advantage, who wagers, that i of the 32 
 different Throws of the Royal-Oak Lottery 
 fhall come at once of 20 times, and that 
 he has fome advantage, who wagers on 22 
 times, fo the neareft to Equality is on 21 
 times. But it muft be remembred, that 
 I have fuppos'd in the former Calculation, 
 the Ball in the Royal-Oak Lottery -to be re- 
 gular, tho' it can never be exactly fo ; for 
 he, who has the fmalleft Skill in Geometry, 
 knows, that there can be no regular Body 
 of 32 fides ; and yet this can be of no Ad- 
 vantage to him who keeps it.
 
 HAZARDS of Game. 31 
 
 To find the Value of the Throws of 
 Dice, as to the Quantity. 
 
 . : 
 
 O thing is more eafy, than by the 
 former Method to determine the Va- 
 lue of any number of Throws of any num- 
 ber of Dice i for in one throw of a Die^ 
 I have an equal Chance for i, 2, 3, 4, 5, 6, 
 confequently my Hazard is worth their 
 Sum 21 divided by their Number 6, that 
 is Si"- Now if one throw of a Die be 
 worth 37, then two throws of a Die, or 
 one throw of. two Dice is worth 7, two 
 throws of two Dice, or one throw of 
 four Dice is worth 14, &c. The general 
 Rule being to multiply the Number of 
 Dice, the Number of Throws, and 3-;- 
 continually. 
 
 This is not to be underftood as if it were 
 an equal wager to throw 7, or above it, 
 with two Dice at one throw ; for he who 
 undertakes to do fo, has the Advantage by 
 21 againft 15. The meaning is only, if 
 I were to have a Guinea, a Shilling, or 
 any thing elfe, for every Point that I threw 
 
 with
 
 3 i Sol? & ion of the 
 
 with two Dice at one throw, my Hazard 
 is worth 7 of thefe, becaufe he who gave 
 me 7- for it, would have an equal probabi- 
 lity of gaining or loling by it, the Chances 
 of the Throws above 7, being as many as 
 of thefe below it : So it is more than an 
 equal wager to throw 14 at leaft at two 
 Throws of two Dice, becaufe it is more 
 probable that 14 will come, than any one 
 number befides, and as probable that it 
 will be above it as below it -, but if one 
 were to buy this Hazard at the rate above- 
 mention'd, he ought juft to give 14 for it. 
 The equal wager in one Throw of two 
 Dice, is to throw 7 at leaft one time, and 
 8 at leaft another time, and fo per vices : 
 The reafon is, becaufe in the firft Cafe I 
 have 2 1 Chances againft 1 5, and in the fe* 
 cond 15 Chances againft 21.
 
 HAZARDS of Game. 33 
 
 Of RAF F LING. 
 
 IN Raffling, the different Throws- and 
 their Chances are thefe ; Where it is 
 to be obferved, that of the 216 ,, r , 
 
 j-rr -T.I r i T^- Throws. Chan. 
 
 different Throws of three Dice, 
 there are only 96 that give Dou- 
 
 blets, or two, at leaft, of a 
 
 3 
 4 
 
 6 
 
 \ 
 
 9 
 
 10 
 
 18 
 
 I 
 
 J 7 
 
 3 
 
 16 
 
 6 
 
 15 
 
 4 
 
 H 
 
 9 
 
 i3 
 
 9 
 
 12 
 
 7 
 
 II 
 
 9 
 
 kind ; fo it is 4 to 5 that with 
 three Dice you mall throw Dou- 
 blets, and it is i to 35 that you 
 throw a Raffle, or all three of 
 a kind. It is evident likewife, 
 that it is an even wager to throw 
 1 1 or above it, becaufe there are as many 
 Chances for 1 1, and the Throws above it, 
 as for the Throws below it ; but tho' it be 
 an even wager to throw 1 1 at one Throw, it 
 is a difad vantage to wager to throw 22 at two 
 Throws, and far more to wager to throw 
 33 at three Throws ; and yet it is more 
 than an equal Wager that you mall throw 
 21 at two Throws in Raffling, becaufe it 
 is as probable that you will, as that you will 
 not throw n, a$ leaft, the firft time, an4 
 F mor
 
 34- Solution of the 
 
 more than probable that you will throw 10, 
 at leaft, the fecond time. 
 
 For an inftance of the plainnefs of the 
 preceding Method, I will mew, how by 
 iimple Subtraction, the moft part of the 
 former Problems may be folv'd. 
 
 Suppofe A and B, playing together, each 
 of them flakes 32 Shillings, and that A 
 wants one Game of the number agreed on, 
 and B wants two ; to find the {hare of the 
 Stakes due to each of them. It's plain, if 
 A wins the next Game, he has the whole 
 64 Shillings j if B wins it, then their 
 Shares are equal ; therefore fays A to B, 
 If you will break off the Game, give me 
 32, which I am fore of, whether I win or 
 lofe the next Game ; and fince you will 
 not venture for the other 32, let us part 
 them equally, that is, give me 16, which, 
 with the former 32 make 48, leaving 16 
 to you. 
 
 Suppofe A wanted one Game, and B 
 three; if A wins the next Game, he has 
 the 64 Shillings; if B wins it, then they 
 are in the condition formerly fuppos'd, in 
 which cafe there is 48 due to A ; there- 
 fore fays A to B, give me the 48 which I 
 am fure of, whether I win or lofe the next 
 Game; and fince you will not hazard for 
 the other 16, let us part them equally, that 
 is, give me S } which, witfc the former 48,
 
 HAZARDS of Gams. 35 
 
 make 56, leaving 8 to you ; and fo all the 
 other Cafes may be folv'd after the fame 
 manner. 
 
 Suppofe A wagers with B, that with 
 one Die he fhall throw 6 at one of three 
 Throws, and that each of them ftakes 108 
 Guineas ; to find what is the proportion of 
 their Hazards. Now there being in one 
 Throw of a Die but one Chance for 6, and 
 five Chances againft it, one Throw for 6 
 is worth 4. of the Stake ; therefore fays B 
 to A y of the 216 Guineas take a fixth part 
 for your fir ft Throw, that is, 363 for your 
 next Throw take a fixth part of the re- 
 maining 1 80, that is, 30 > and for your 
 third Throw, take a fixth part of the re- 
 maining 150, that is, 25, which in all 
 make 91, leaving to me 125; fo his ha- 
 zard who undertakes to throw 6 at one of 
 three Throws, is 91 to 125. 
 
 Suppofe A had undertaken to throw 6 
 with one Die at one Throw of four, and 
 that the whole Stake is 1296 ; fays A to B y 
 every Throw for 6 of one Die, is worth 
 the fixth part of what I throw for ; there- 
 fore for my firft Throw give me 2 16, which 
 is the fixth part of 1296, and there re- 
 mains 1080, I muft have the fixth part 
 of that, viz. 1 80, for my fecond Throw ; 
 and the fixth part of the remaining 900, 
 which is 1 50, for my third Throw j and 
 F 2 the
 
 36 Solution of the 
 
 the fixth part of the laft remainder 750, 
 which is 1 2 5 for my fourth j all this addedt 
 together makes 671, and there remains to 
 you 625 j fo it is evident, that As Hazard, 
 in this Cafe, is to 5's 671 to 625. 
 
 Suppofe A is to win the Stakes (which 
 we fhall fuppofe to be 36) if he throws 7 
 at once of twice with two Dice, and B is 
 to have them if he does not j fays B to A y 
 the Chances which give 7 are 6 of the 36, 
 which is as much as i of 6 5 therefore for 
 your fiift Throw you mall have a fixth 
 part of the 36, which is 6 ; and for your 
 next Throw a fixth part of the remainder 
 30, which is 5 5 this in all makes n ; fo 
 you leave 25 to me j fo A's Hazard is to 
 J5's as n to 25. 
 
 It were cafy," at this rate, to calculate 
 the moft intricate Hazards, were it not that 
 Fractions will occur j which, if they be 
 more than 4, may be fuppos'd equal to an 
 Unit, without caufing any remarkable Er- 
 ror in great Numbers. 
 
 It will not be amifs, before I conclude, 
 to give you a Rule for finding in any num- 
 ber of Games the Value of the firft, becaufe 
 Huygens's Method, in that cafe, is fomething 
 tedious. 
 
 Suppofe A and B had agreed, that he 
 
 mould have the Stakes who did win the 
 
 x firft 9 "Games, and A had already won one 
 
 * of
 
 HAZARDS of Game. 37 
 
 of the 9 ; I would know what mare of .B's 
 Money is due to A for the Advantage of* 
 this Game. To find this, take the firft 
 eight even Numbers 2, 4, 6, 8, 10, 12, 14, 
 j 6, and multiply them continually , that 
 is, the firft by the fecond, the product by 
 the third, GV. take the firft eight odd Num- 
 bers i, 3, 5, 7, 9, n, 13, 15, and do juft 
 fo by them, the produc~l of the even Num- 
 ber is the Denominator, and the produd: of 
 the odd Number the Numerator of a Frac- 
 tion, which exprefleth the quantity of B's 
 Money due to A upon the winning of the 
 firfl Game of 9 ; that is, if each ftak'd a 
 number of Guineas, or Shillings, 6?r. ex- 
 prefs'd by the produdt of the even Num- 
 bers, there would belong to A, of jB's 
 Money, the Number exprefs'd by the pro- 
 duct of the odd Numbers. For Example, 
 Suppofe A had gain'd one Game of 4, then 
 by this Rule, I take the three firfl even 
 Numbers, 2, 4, 6, and multiply them con- 
 tinually, which make 48, and the firft 
 three odd Numbers, i, 3, 5, and multiply 
 them continually, which make 1 5 ; fo there 
 belongs to A JLJ, of B's Money, that is, if 
 each ftak'd 48, there would belong to A> 
 befides his own, 1 5 of B's. Now by Huy- 
 gem's Method, if A wants but three Games 
 while B wants four, there is due to A -|4 
 of the Stake; by this Rule there is due 
 
 to
 
 38 Solution of the 
 
 to A 4| of 's Money, which is ~| of the 
 Stake, which, with his own j. of the 
 Stake, makes |. or .11 of the Stake; and 
 fo in every Caie you will find Huygens's 
 Method, and this will give you the fame 
 Number : A Demonstration of it you may 
 
 tin a Letter of Monfieur Pajcal's to 
 Dnfieur Fermat ; though it be otherwife 
 exprefs'd there than here, yet the confe- 
 quence is eafily fupply'd. To prevent the 
 labour of Calculation, I have fubjoin'd the 
 following Table, which is calculated for two 
 Gameflers, as Monfieur Huygens's is fot 
 three. 
 
 If each of us flake 256 Guineas in 
 
 96 
 
 V o 
 
 3 
 
 - 
 
 6 
 
 5 
 
 4 
 
 Game 
 
 63 
 
 70 
 
 80 
 
 2 ift 
 
 Games 
 
 126 
 
 140 
 
 1 60 
 
 3 \ft 
 Games 
 
 Games 
 
 182 
 224 
 
 200 
 240 
 
 224 
 
 256 
 
 5 l ft 
 Games 
 
 248 
 
 256 
 
 
 6 iji 
 Games 
 
 256 
 
 
 
 192 
 
 256 
 
 2 
 
 128 
 
 I 
 
 256 
 
 2 5 6 
 
 
 
 The
 
 HAZARDS of G$me. 3 9 
 
 The Ufe of the Table is plain ; for let 
 our Stakes be what they will, I can find 
 the portion due to me upon the winning 
 the firft, or the firft two Games, Gfr. of 
 2, 3,4, 5, 6. For Example , If each of us 
 had ftak'd 4 Guineas, and the number of 
 Games to be play'd were 3, of which I 
 had gain'd i, fay, As 256 is to 96, fo is 4 
 to a fourth. 
 
 256 : 96 :: 4 ; if. 
 
 'To find what is the Value of hh Hazard, 
 who undertakes, at the firft Throw, tQ caft 
 Doublets, in any given number of Dice. 
 
 In two Dice it is plain, to avoid Dou- 
 blets, every one of the fix different Throws 
 of the firft, can only be combin'd with 
 five of the fecond, becaufe one of the fix 
 is of the fame kind, and confequently 
 makes Doublets j for the fame reafon, the 
 thirty Throws of two Dice, which are 
 not Doublets, can only be combin'd with 
 four Throws of a third Dice, and three 
 Throws of a fourth Dicej fo generally 
 it is this Series, 
 
 6x5x4x3x2x1x0, 
 6x6x6x6x6x6x6, 
 
 The.
 
 40 Solution of the 
 
 The fecond Series is the Sum of the 
 Chances, and the firfl the Number of 
 Chances againft him who undertakes to 
 throw Doublets, each Series to be conti- 
 nu'd fo many terms, as are the number -of 
 Dice. For Example, If one mould under- 
 take to throw Doublets at the firft Throw 
 of four Dice, his Adverfary's Hazard is 
 
 *** x n = , or \ , leaving to him 
 
 6x6x6x6 1296 1 8 * 
 
 44, fo he has x 3 to 5. In feven Dice, you 
 fee the Chances againft him are o, be- 
 caufe then there muft neceffarily be Dou- 
 blets.
 
 HAZARDS of Game* 41 
 
 
 Of W H i s T. 
 
 If there be four playing at Whift, it is 15 
 to i that any two of them /ball not have the 
 four Honours, which I demonftrate thus. 
 
 SUppofe the four Gamefters be A, B, C, 
 D : \iA and B had, while the Cards are 
 a dealing, already got three Honours, and 
 Wanted only one, lince it is as probable 
 that C and D will have the next Honour, 
 as A and E ; if A and B had laid a Wager 
 to have it, there is due to them but | of 
 the Stake : If A and B wanted two of the 
 four, and had wager'd to have both thofe 
 two, then they have an equal hazard to get 
 nothing, if they mifs the firfl of thefe two, 
 or to put themfelves in the former Cafe 
 if they get j fo they have an equal Ha- 
 zard to get nothing, or -J, which, by 
 Prop. i. is worth * of the Stake ; fo if they 
 want three Honours, you will find due to 
 them 4- of the Stake j and if they wanted 
 four, -rr f ^ e Stake, leaving C and D 
 4^; foC and> can wager 15 to i, that 
 G A
 
 42 Solution of the 
 
 A and B mall not have all the four Ho- 
 nours. 
 
 It is n to $ that A and B Jhall not have 
 three of the four Honours^ which I prove 
 thus : 
 
 It is an even Wager, if there were but 
 three Honours, that A and B mall have 
 two of thefe three, fince 'tis as probable 
 that they will have two of the three, as 
 that C and JD (hall have them ; confe- 
 quently, if A and B had laid a Wager to 
 have two of three, there is due to them 1 
 of the Stake. Now fuppofe A and B had 
 wager'd to have three of four, they have 
 an equal hazard to get the firft of the 
 four, or mifs it ; if they get it, then they 
 want two of the three, and confequently 
 there is due to them ! of the Stake j if 
 they mifs it, then they want three of the 
 three, and confequently there is due to 
 them f of the Stake j therefore, by Prop. 
 i. their Hazard is worth T s r , leaving to C 
 and D -K.. 
 
 A and B playing at Whift againft G and D; 
 A and B have eight of ten, and C and T) 
 nine, and therefore can't reckon Honours j to 
 find the proportion of their Hazards. 
 
 There
 
 HAZARDS of Game. 43 
 
 There is T s r due to A and B upon their 
 hazard of having three of four Honours > 
 but fince C and D want but one Game, 
 and A and $ two, there is due to A and 
 j5 but -y, or T ^ more upon that account, 
 by Prop. 4. this in all makes T 9 r , leaving to 
 C, and D T T T j fo the hazard of A and 5 to 
 that of C and Z), is as 9 to 7. 
 
 In the former Calculations I have ab- 
 ftra&ed from the fmail difference of having 
 the Deal, and being Seniors. 
 
 All the former Cafes can be calculated 
 by the Theorems laid down by Monfieur 
 Huygens $ but Cafes more compos'd re- 
 quire other Principles : for the eafy and 
 ready Computation of which, I mall add 
 pjie Theorem more, demonftrated after Mon- 
 fieur Huygens's Method. 
 
 TH EORE M. 
 
 If I have p Chances for a, q Chances 
 for b t and r Chances for c, then my hazard 
 
 is worth ap+ q ^ cr ; that is, a multiplied 
 
 /-t-?-H r 
 into the number of its Chances added to 
 
 b, multiplied into the number of its Chan- 
 ces, added to c, multiplied into the num- 
 ber of its Chances, and the Sum divided by 
 the Sum of Chances of a, b t c. 
 
 G 2 To
 
 ^4 Solution of the 
 
 To inveftigate as well ,as demonn;ra.te 
 this Theorem, fuppcfe the value of my ha- 
 zard be x, then x muft be fuch, as having 
 it, I am able to purchafe as good a hazard 
 again in a juft and equal Game. Suppofe 
 the Law of it be this, That playing with 
 fo many Gamefters as, with myfelf, make 
 up the number pj-q-\-r t with as many of 
 them as the number p reprefents, I make 
 this bargain, that whoever of them wins 
 mall give me a, and that I (hall do fo to 
 each of them if I win j with the Game- 
 fters reprefented by the number of q y I 
 bargain to get ^, if any of them win, and 
 tp give b to each of them, if I win my 
 felf j and with the reft of the Gamefters, 
 whofe number is r< i, I bargain to give, 
 or to get c after the fame manner ; Now 
 all being in an equal probability to gain, I 
 Jiave^ Chances to get a y q Chances to get 
 b, and r- i Chances to get r,-and one 
 Chance, viz. when I win myfelf, to get 
 fx -f- qx -+-rx ap bq re -h c , which, 
 if it be ibppos'd equal to c , then I have p 
 Chances for a, q Chances for b t and r 
 Chances for c (for I had juft now r i 
 Chances for it) and therefore, in cafe px -f- 
 qx -+- rx ap bq rc-i-c = c, then is 
 
 ... */. 4- fy 4- t r 
 
 By
 
 HAZARDS of Game. 45 
 
 By the fame way of reafoning ou will 
 find, If I have p Chances for </, q Chances 
 for b, r Chances for c, and s Chances for 
 
 d, that my hazard is 
 
 In NUMBERS. 
 
 If I had two Chances for 3 Shillings, 
 four Chances for 5 Shillings, and one 
 Chance for 9 Shillings, then, by this Rule, 
 my hazard is worth 5 Shillings j for 
 
 _^ s to 
 
 prove, that with five Shillings I can pur- 
 chafe a like hazard again j for fuppofe I 
 play with fix others, each of us flaking 
 5 Shillings ; with two of them I bargain, 
 that if either of them win, he muft give 
 me 3 Shillings, and that I mail do fo to 
 them ; and with the other four I bargain 
 juft fo, to give or to get 5 Shillings : 
 This is a juft Game, and all being in an 
 equal probability to win 5 by this means 
 I have two Chances to get 3 Shillings, 
 four 'Chances to get 5 Shillings, and one 
 Chance to get 9 Shillings, viz. when I 
 win my felf ; for then out of the Stake, 
 which makes 35 Shillings, I muft give the. 
 firft two 6 Shillings, and the other four 20 
 Shillings, fo there remains juft 9 to myfelf.
 
 46 Solution of the 
 
 It is eafy, by the help of this Theorem^ 
 to calculate in the Game of Dice, com- 
 monly call'd Hazard, what Mains are beft 
 to fet on, and who has the Advantage, 
 the Carter or Setter. The Scheme of the 
 Game, as I take it, is thus } 
 
 Mains. 
 V. 
 VI. 
 VII. 
 VIII. 
 IX. 
 
 Throws next following for 
 
 -A. 
 
 The Coffer. 
 V. 
 VI. XII. 
 VII. XL 
 VIII. XII. 
 IX. 
 
 The Setter. 
 II. III. XL XII. 
 XL II. III. 
 XII. II. III. 
 XL II. III. 
 II. III. XL XII. 
 
 By an eafy Calculation you will find, 
 if the Carter has IV,. and the Setter VII, 
 there is due to the Carter JL of the Stake ; 
 if he has 
 
 V againrt VII, ^ of the Stake. 
 
 VI againft VII, T s - of the Stake. 
 
 IV againrt VI, 4 of the Stake. 
 
 V againft VI, *. of the Stake. 
 IV againft V, f of the Stake. 
 
 I need not tell the Reader, that IV is 
 
 the fame with X, 
 with VIII. 
 
 V with IX, and VI 
 Suppofe
 
 HAZARDS of Game. 47 
 
 Suppofe then VII be the Main : To find 
 the proportion of the hazard of the Cafter 
 to that of the Setter. 
 
 By the Law of the Game, the Carter, 
 before he throws next, has four Chances 
 for nothing, viz. thefe II, III, XII ; eight 
 Chances for the whole Stake, viz. thofe 
 of VII, XI ; fix Chances for I., viz. thofe 
 IV, X ; eight Chances for -J-, viz. thofe 
 of V, IX -, and ten Chances for T S T , viz. 
 thefe of VI, VIII -, fo his hazard, by the 
 preceding Theorem, is 
 
 4x0-4-8x1 + 6x j- -4- 8 x-J- -f- 10 x T V 
 ~ 
 
 Now to fave the trouble of a tedious 
 reduction, fuppofe the Stake which they 
 play for be 36, that is, the Setter had laid 
 down 1 8 ; in that cafe, every one of 
 thefe Fractions are fo many parts of an 
 Unit/ which being gather'd into one Sum, 
 give 17*4 to the Cafter, leaving 1814 to 
 the Setter ; fo the hazard of the Cafter is 
 to that of the Setter 244, 251. 
 
 Suppofe VI, or VIII, be the Main, then 
 the Share of the Cafter is 
 
 JI. 
 
 III. VI. IV. V. 
 
 XL XII. X. IX. VIII. VII. 
 
 leaving
 
 48 Solution of the 
 
 leaving to the Setter 18^. ; fo the hazard 
 of the Carter is to that of the Setter as' 
 6961 to 7295. 
 
 Suppofe V, or IX be the Main, then the 
 Share of the Carter is 
 
 II. 
 III. 
 
 XI. IV. VI. 
 
 XII. V. X. IX. VIII. VII. 
 
 6xo-f 4x i+6x 
 
 leaving to the Setter i8 T VV > f the hazard 
 of the Carter is to that of the Setter as 
 1396 to 1439. 
 
 It is plain, that in every cafe the Ca- 
 rter has the Difadvantage, and that V, or 
 IX, are better Mains to fet on than VII, 
 becaufe, in this laft Cart, the Setter has 
 but 1 8 and -J4> or jVo-J whereas, when V 
 or IX is the Main, he has 183- ; likewise 
 VI, or VIII, are better Mains than V, or 
 IX, becaufe .J-yJ. is a greater Fraction than 
 
 TTT- 
 
 All thofe Problems fuppofe Chances, 
 which are in an equal probability to hap- 
 pen > if it fhould be fuppos'd otherwilc, 
 there will arife variety of Cafes of a quite 
 
 different
 
 HAZARDS of Game. 49 
 
 different nature, which, perhaps, 'twere not 
 unpleafant to conlider : I ihall add one Pro- 
 blem of that kind, leaving the Solution to 
 thofe who think it merits their pains. 
 
 In Parattelipipedo cujus laterafunt ad in- 
 i)icem in ratione a, b, c : Invenire quota vice 
 quivisfufcipere poteft, ut datum quodvis fla- 
 num, v.g. nbjaciaf.
 
 A 
 
 DEMONSTRATION 
 
 O F T H E 4 
 
 Gain of the BANKER 
 
 In any Circumftarice of the 
 
 Game call'd Pharaon* 
 
 And how to determine the ' 
 
 Odds at the ACE of HEARTS, 
 or FAIR CHANCE ; 
 
 With the ARITHMETICAL SOLUTION of 
 fome Queftions relating to LOTTERIES: 
 
 -And a few Remarks upon HAZARD and 
 BACKGAMMON,
 
 L S3 J 
 
 
 Of the Game of PH A R A o N. 
 
 N order to demonftrate the Odds 
 or Gain of the Banker in any 
 circumftance of the Game of 
 Pharawi, it will be neceflary to 
 folve the following Problem. 
 
 P R O B. I. 
 
 Two Perfons A, B, out of a heap of 9 Cards, 
 four of which are red and five black, un- 
 dertake to draw a red one blindfold^ ' and 
 he fiall be reputed to win that draws the 
 firft : Now fuppofmg A to have the firft 
 choice,, B the jecond, A the third, and Jo 
 on by turns till one of them wins ; Qusere, 
 'The proportion of their Hazards ? 
 
 SOL u TION. 
 
 Let n be the number of ail the Cards, r 
 the number of red ones, b the number of 
 
 black
 
 54 Of the Game 
 
 black ones, and i the whole Stake, or the 
 Sum play'd for. 
 
 i, Since A has r Chances for a red Card, 
 and b Chances for a black one, it follows 
 by the third Propofition that his Expe&a- 
 
 tion is worth -^ b or of the Stake, e- 
 
 qual to x i ; and accordingly let it be 
 
 agreed between the two Gamefters, that 
 inftead of yfs attempting to draw a red 
 Card he mall actually take out a black one, 
 
 and as an equivalent mall have -^ paid him 
 out of the Stake, which being done, there 
 will remain i = ^^ =- . 
 
 n n n 
 
 2, Since the remaining Cards are # r, 
 and B has r Chances for a red Card, it fol- 
 lows that his Probability of winning will be 
 
 -^ , and confequently his Expectation 
 upon the remaining Stake will be -~ 
 x = - Butinfleadof B'sdraw.- 
 
 n n x n I 
 
 ing, we will fuppofe the Sum ny j n ^ l paid 
 him out of the Stake, and that a black 
 Card being taken out of the heap as before, 
 
 then the remaining Stake will be ^ 
 
 rb bxn i b x n b bxb i 
 
 X I X i """ X I """"*" x n l * 
 
 3;
 
 of PHARAON. 55 
 
 3% Now it comes to 4's turn again to 
 chufe ; and fince he has ftill r Chances for 
 a red Card, and the number of remaining 
 Cards are n 2, his Probability of winning 
 
 this time will be - , and his Expectation 
 upon the remaining Stake a * a ~| will be 
 
 ~* ~ ' * ~r > which we will likewife 
 fuppofe to be paid him out of the Stake, 
 and then there will remain ""?. --. 
 
 n-n.ii l 
 3x3 i x r 3x3 jx 2 
 
 xg ^ , or nxn _ l x a " 
 
 b X b IX b bxb ix3 2 c 
 
 == - OUt Of 
 
 X a I X 2 - X I X 2 7 
 
 1-1 T-. 1 3x3 1x3 2Xr 
 
 which B may have 
 
 IX 2X 3 
 
 and fo we may proceed till the whole Stake 
 is exhaufted. 
 
 But I hope what has been faid is fufficient 
 to fee the Law of Continuation, and form 
 
 this general Series ; viz. - -f- - P -j- 
 
 + :S,r. In which 
 'tis evident that P, Q^, R, S, &c. denote 
 the preceding Terms. Now to apply this 
 Series to practice, we muft take as many 
 Terms of it, as there are Units in b -\- i ; 
 for fince reprefents, the number of black 
 Cards, the number of Drawings cannot 
 exceed b -J- 1 3 therefore take for y^the firft,
 
 56 Of tie Game 
 
 third, fifth, &c. Terms ; and for B, rfie 
 fecond, fourth, fixth, &c . Terms, and the 
 Sum of thole Terms will he the refpeclive 
 Expectations of ^, B$ or becanfe the Stake 
 is rix'd, thefe Sums will be proportional to 
 their refpedivc Probabilities of winning. 
 For inftance, if we apply this to the pre- 
 fent Cafe, the general Series will be -f- 
 4? H- fQ^r- yR.-f- yS -f- iT. And to 
 bring this Series into whale Numbers, let 
 us aSurne x a whole Number, which mul- 
 tiply'd by % ihall be a whole Number =3 P. 
 
 Therefore fmce ^ = P, ^ x-J-s=Q^an4 
 
 ~ x 4 x f = R, &c. each of which Terms 
 are to be whole Numbers: Now 'tis evi- 
 dent that the Denominator of the Fraction 
 reprefented by P, is an aliquot part of that 
 reprefented by Q5 and every Denomina- 
 tor an aliquot part of the next following, 
 and fo the laft Denominator 9x8x7 x6x 
 5 x 4 is a Multiple to all the preceding De- 
 nominators; and confequently if each Nu- 
 merator be multiplied by it, inftead of x y 
 and the Produces divided by their refpective 
 Denominators, the Quotients will all be 
 whole Numbers, this is univerfally true : 
 But in this Cafe, if the laft Fraction 
 
 9x8x7x6x5x4 
 
 9/2. 7?6 3 * ts Denominator 126 is the leaft 
 
 common, 
 
 1
 
 0/PHARAON. 57 
 
 common Multiple to all the reft of the 
 Fractions when reduc'd to their leaft Terms. 
 
 Which being determin'd, the Terms of 
 the Series are eafily found by the following 
 Operation to be 56 -f- 35 -4- 20 4- 10 -4- 
 4-M- 
 
 OPE RATION. 
 
 126 20 = R 
 
 504 60 
 
 9 ) - 6) 
 
 56=? 10 = 8 
 
 280 ~20~ 
 
 8) - 5) 
 
 4 i 
 
 140 4 
 
 7) - 4) 
 
 20 == R i =s U 
 
 Wherefore aligning to ^ 56-4-20-4-4 = 80 
 And toB -------- 35^10+1 = 46 
 
 And their Probabilities of winning will be 
 as 80 to 46, or as 40 to 23. And if there 
 be never fo many Gamefters A, B, C, D, 
 &c. the Probabilities of winning may 'as 
 eafily be affign'd by the general Series, as 
 for thefe two. 
 
 1 REMARK^
 
 Of the Game 
 
 REMARK. 
 
 The preceding Series may in any par- 
 ticular Cafe be eafily fhorten'd; for if r=i, 
 then the Series will be 
 
 ' 
 
 If r = 2, then the Series will be - 
 
 n x n I 
 
 x -i -+- n 2 -+- n 3 -f- n 4, &c. 
 If r=^, then the Series will be 
 
 X .. --: X 2 
 
 x n i xw 2 -j-# 2 x 3 4-# 3 x 4 
 &c. 
 
 If r = 4, then the Series will be 
 
 1 X n ixa 2X/; 3 -J- 2X jX/z 4. 
 
 &C. 
 
 Wherefore rejecting the common Multi- 
 plica tors, the feveral Terms of thofe Series 
 taken in due order will be proportional to 
 the feveral Expectations of any number of 
 Gamefters. Thus in the Cafe of this Pro- 
 blem, where 72=9, ^ = 4, =5, the 
 Terms of the Series will be 
 
 For A. For B. 
 
 8x7x6 = 336 7x6x5 = 210 
 
 6x5x4 = ' 20 5x4x3 = 60 
 
 4x3x2 = 24 3x2x1 = 6 
 
 480 276 
 
 Hence
 
 of PHARAON. 59 
 
 Hence it follows, that the Probabilities 
 of winning will be refpe&ively as 480 to 
 276, or dividing both by 12, as 40 to 23, 
 the fame as before, 
 
 The Game of PHARAON. 
 
 Rules of the Play. 
 
 Firft, The Banker holds a Pack of 52 
 Cards. 
 
 Secondly, He draws the Cards one after 
 the other, laying them alternately to his 
 right and left-hand. 
 
 Thirdly, The Ponte may, at his choice, 1 
 fet one or more Stakes, upon one or more 
 Cards taken out of his parcel of 13 Cards, 
 from the Ace to the King inclufive, call'd 
 a Book* either before the Banker has begun 
 to draw the Cards, or after he has drawn 
 any number of Couples, which are com- 
 monly call'd Pulls. 
 
 Fourthly, The Banker wins the Stake of 
 the Ponte, when the Card of the Ponte 
 comes out in an odd place on his right- 
 hand ; but lofes as much to the Ponte 
 when it comes out in an even place on his 
 left-hand. 
 
 Fifthly, The Banker wins half the Ponte's 
 Stake, when in the fame pull the Card of 
 the Ponte comes out twice. 
 
 I 2 Sixthly,
 
 60 Of the Game 
 
 Sixthly, When the Card of the Ponte, 
 being once in the Stock, happens to be the 
 Lift, the Ponte neither wins nor lofes. 
 
 Seventhly, The Card of the Ponte being 
 but twice in the Stock, and the two laft 
 Cards happening to be his Cards, he then 
 lofes his whole Stake, 
 
 PR OB. II. 
 
 $0 find at Pharaon the Gain of the Banker > 
 when any number of Cards remain in 
 the Stock ; having the number of times that 
 the Ponte 's Card is contain d in it, given. 
 
 WHerefore this Problem admits of four 
 Cafes, fince the Ponte's Card may 
 be contain'd either once, twice, thrice, or 
 four times in the Stock, 
 
 Solution of the firfr, CASE. 
 
 The Banker, according to the Law of 
 the Game, has the following number of 
 Chances for winning and lofmg ; viz. 
 
 i Chance for winning i 
 i Chance for loiing i 
 i Chance for winning i 
 i Chance for lofmg i 
 i Chance for winning i 
 i Chance for lofmg o 
 
 
 Hence
 
 of PHARAON* 6* 
 
 Hence the Banker has one Chance more 
 for winning than for lofing, and the num- 
 ber of all the Chances are equal to , the 
 number of Cards in the Stock j therefore -1 
 33 the Gain of the Banker upon the Stake. 
 To illuftrate this Cafe, 
 
 Let it be required to find the Gain of 
 the Banker, when there are 20 Cards re- 
 maining in the Stock, and the Ponte's Card 
 
 but once in it, =3 
 
 n 20 
 
 He gains the twentieth part of 
 his Stake. 
 
 Solution of the fecond CASE. 
 
 By the Remark belonging to the pre- 
 ceding Problem, it appears, that the Chances 
 which the Banker has to win or lofe his 
 Stake would be proportional to thefe Num- 
 bers n i, n 2, n 3, &G. were the 
 Banker not aliow'd half the Stake upon 
 drawing of Doublets, or the Ponte's Card 
 twice together ; upon which account the 
 number of Chances reprefented for the Ban- 
 ker by i for winning, mufl be divided 
 into two parts 2, and i, whereof the 
 firil is proportional to the probability which 
 the Banker has for winning the whole Stake 
 of the Ponte, and the fecond is proportional 
 to the probability of winning the half of it: 
 for fmce the Banker is not intitled to the 
 i whole
 
 62 Of the Game 
 
 whole Stake upon the Ponte's Card coming 
 out in an odd place, till he knows whether 
 the next Card be the Ponte's, his Chance 
 for winning the whole Stake can be no 
 greater than the Ponte's ; becaufe there is 
 the fame reafon, and eafily prov'd, that the 
 fecond Card, and not the firft, mould be 
 the Ponte's, as the firft Card, and not the 
 fecond. Wherefore the Chances that the 
 Banker and the Ponte have to win the whole 
 Stake are equal, with refpect to the order 
 of drawing, and confequently mould be ex- 
 prefs'd by the fame number 3 whence arifcs 
 this Scheme for determining the .number of 
 Chances which the Banker has for win- 
 ning and lofing the Stake, and the num- 
 ber of Chances for winning the half Stake, 
 jeprefented by_y. 
 
 SCHEME. 
 
 \ ~" Chances for winning \ 
 n 2 Chances for lofing i 
 
 4 ^ J-Chances for winning^ 
 n 4 Chances for lofing i 
 
 ^-Chances for winning^ I 
 n 6 Chances for lofing i 
 
 i Chance for winning i 
 
 Firft,
 
 Of PH ARAON. 63 
 
 Firft, $r^r- = Number of Chances for 
 gaming y from the nature of the Scheme, 
 and n x ^^ = Sum of all the Chances. 
 Therefore -" ~~ 2 - is the probability of the 
 
 . r X I J 
 
 Banker's gaining^, and n ^ n __ l x7=the 
 Banker's Gain upon y. 
 
 Secondly, The Banker has but i Chance 
 more to win the Stake than to lofe ; the reft 
 for winning and lofmg being equal deftroy 
 
 each other. Therefore - is the pro- 
 
 /; l A 
 
 n x 
 
 2 
 
 bability of getting it, and - ~ x i = 
 
 n x 
 
 VJ ff i Z 
 
 - - is his Gain upon the Stake. 
 
 a x n i 
 
 Therefore the Banker's Gain vipon the 
 
 i i 2 2 ti '2. X -, -4- 2 
 
 whole is - - ' 
 
 *" -r- ' -r i 
 
 MX ! j li y f 
 
 To illuftrate this Cafe by an Example in 
 Figures, Let it be required to find the Gain 
 of the Banker when there are 2d Cards 
 remaining in the Stock, and the Ponte's 
 Card twice in it. 
 
 fe = :-^=TVV = y T nearly. 
 
 Anfaer, About the thirty-fourth part of 
 his Stake. 
 
 That
 
 64 Of tie Game 
 
 That x ^ J- is the Sum of all the 
 
 2 
 
 Chances, may be thus prov'd. 'Tis evi- 
 dent by infpection of this Scheme, that 
 they are equal to this Series n I, n 2, 
 n __ o n 4, &c. And - x 
 
 ^ ? K x a i 
 
 i -f- 2 -h n 3, &c. is a Series- 
 belonging to the preceding Problem, which 
 exprefles the Sum of the Probabilities of 
 winning, which belong to the two Game- 
 flers, when the number of all the Cards is 
 72, and the number of red ones two. It 
 therefore exprefles like wife the Sum of the 
 Probabilities of winning which belong to 
 the Ponte and Banker in the prefent Cafe, 
 there being two of the Ponte's Cards in 
 the Stock ; but the Sum of thefe Probabili- 
 ties of winning are equal to Unity, becaufe 
 the Numerators of thofe Fractions which 
 exprefs their refpective Probabilities of win- 
 ning being added together, is equal to the 
 common Denominator, and fo equal to ah 
 Unit, confequently 
 
 x l_ , x i + 2 -f- 3, &c. 
 is equal to an Unit ; wherefore putting 
 /= i, n 2, n 3, &c. we mail 
 have -^i x/= i, and therefore /= 
 
 H^ I 
 
 x "T"' 
 
 Solution
 
 65 
 
 Solution of the third CASE. 
 
 By the preceding Problem, the number of 
 Chances the Banker has for winning or lofing 
 the Stake would be proportional to thefe 
 Numbers n i x ;; 2, n 2 x # 3, 
 &c. but that on account of the Doublets, 
 the Banker's Chances for winning, as in the 
 preceding Cafe, are divided into two parts : 
 For inftance, n i x # 2 is divided imo 
 the parts n 2 x n 3, and 2 x n 2 ; 
 the former of which is proportional to the 
 Chances that both Banker and Ponte have 
 to win the whole Stake the firft pull, be- 
 caufe there is the fame probability that the 
 Ponte's Card mould be drawn the fecond 
 time and mifs'd the firft, as drawn the firft 
 time and mifs'd the fecond ; and the latter 
 part, viz. 2 x n 2 is proportional to the 
 Chances for both firft and fecond Cards be- 
 ing the Ponte's the firft pull : and as 2 x/z 2 
 is eafieft found by fubtracling n 2 xn 3 
 from ;/ i x n 2 ; fo by fubtracting 
 n 4 x n 5, the number of Chances that 
 the Banker and Ponte each have, for winning 
 the whole Stake the next pull from n i 
 x n 4, there will remain 2x;z 4, the 
 number of Chances for the third and fourth 
 Card being the Ponte's, and fo for the reft : 
 whence is eafily deduc'd the Method of 
 K forming
 
 66 Of tie Game 
 
 forming' the following Scheme, which mews 
 what Chances the Banker has for winning 
 and lofing. 
 
 SCH EME. 
 
 2 
 
 f-2x~ 3 ? CI f j Ji 
 
 c 2 x n 23 ly 
 
 n 2 x n 3 Ch. for lofing i 
 
 4*-4*-5lch .for win . 
 C. 2 x n 43 
 
 n 4 x n 5 Ch. for lofing i 
 
 f ; 
 
 i x n C v 
 
 6 x 7 Ch. for lofing i 
 
 Hence it appears, that there are no more 
 Chances for winning the Stake than for lofing 
 it j fo all the Banker's Advantage in this 
 cafe is upon the half Stake, which depends 
 upon drawing of Doublets at one of the 
 Pulls, and all the Chances he has for that 
 may be found thus. 
 
 By the above Scheme, 2 x n 2 is the 
 firft Term of the Series to win v, 4 is the 
 common difference of the reft of "the Terms, 
 and 4 the laft Term j and fo the Sum of 
 all the Terms by the known Laws of Arith- 
 metical Progreflion is " x n ~ 2 , which di- 
 vided
 
 RAON. 67 
 
 vided by the Sum of all the Chances = 
 * x " "~ '- ^-^- by the preceding Problem, 
 
 gives ~~rr~ f r l ^ e probability of win- 
 ning y ; therefore 2 x j*_ t x y is the Ban- 
 ker's Gain, equal - x< 3 _- , fuppofing ^=4. 
 
 EXAMPLE. 
 
 Query, The Banker's Gain when the Stock 
 confifts of 20 Cards, and the Ponte's Card 
 thrice in it ? 
 
 4X i 4 x 19 ~~ fT *" ~ 7T ' 
 
 Anfwer, About the twenty-fifth part of 
 his Stake. 
 
 Solution of the fourth CASE. 
 SCHEME. 
 
 {^ rCh. for win. 
 
 3 x 2 x n 3 3 c y 
 
 2 x 3 x n 4 Ch. for lofing i 
 
 -67 ch f C i 
 
 3 x 4 x 5-) c^ 
 
 4 x 5 x n 6 Ch. for lofing i 
 
 iy 
 
 2 x i x o Chance for lofing 
 
 K^2 HeTe
 
 68 Of the Game 
 
 Here alfo the Chances for winning and 
 lofing the Stakes are equal, fo that the 
 Banker's Gain depends upon y y the half 
 Stake. 
 
 And the Chances to win y are 
 
 13x2x1! r 2 x i 
 
 Equal to<3x4x3> = 3x<4x3 
 
 03x6x5) C 6 * 5 
 
 &c. &c. 
 
 And to find the Sum of thofe Produces, 
 viz. 2x1 -4-4x3 -4- 6x5, &c. continued 
 to any number of Terms, whofe Factors 
 are in Arithmetical Progreffion, we muft 
 premife the following 
 
 LEMMA. 
 
 Subtract the fecond Product from the 
 third, and the third Product from the 
 fourth, and call the Remainers firft Diffe- 
 rences j then fubtract thofe Differences from 
 each other, and call the Remain er a fecond 
 Difference. See the following Scheme. 
 
 2X1= 2 
 
 1* I2 l8 Q 
 
 6x5 = 30 8 
 8 x 7 = 56 
 
 Now if we call the firft Product a, the 
 fecond b, the firft of the firft Differences /, 
 *he fecond Difference /'; and if the num- 
 ber
 
 0/PHARAON. 69 
 
 ber of Products which follow the firft be 
 called x, the Sum of all the Products will 
 
 be equal to a -f- x x I - -f- ' x ^ ^ x / -f- 
 
 -Y X I X 2 ; 
 
 x x x a . 
 
 I 2 3 
 
 Or 
 
 * x 
 
 12 4- -f x ^ x 18 4- 
 
 X 2 Q 
 
 x x 8. 
 
 Or 
 
 2 4- 5-lx 4- 5** 4- i^xxx, which mul- 
 tiplied by 3, the common Multiplier to all 
 the Terms, the Product 6 4- 17*4-; 15** 
 4- 4# 3 will be the Sum of all the Chances 
 for winning of y, whofe component Parts, 
 by Sir IJ'aac Newton's Method of the Inven- 
 tion of Divifors, is readily found to be 
 
 2x -f- 2, 2x -+- 4, and 4Af 3 ^ wherefore 
 
 3 
 
 l2L is the number of 
 
 Chances for winning ofy. Now A; = to the 
 number of Terms to be added, bating i, 
 therefore make #-f- i=/>, equal to the 
 number of Terms, and then the Chances 
 for winning of y are .'/"H-'xiFT fi|Jt 
 
 according to the Scheme it appears, that 
 the number of Terms to be added are equal 
 
 to ^ ^ > wherefore writing ^ ^ for/, 
 
 and
 
 70 Of the Game 
 
 and the Chances for winning are equal to 
 5 , which divided by the Sum 
 of all the Chances, viz. *Z^m-***-3 
 and the Quotient n _^"~ t *_ is the proba- 
 bility of winning y. and ib zn ~~ ; x ^ \* 
 
 J n i x 3 
 
 the Gain of the Banker, or - 2 """" 5 _ 
 
 2 X - IXK - 3 
 
 fuppofing y = 4. 
 
 EXAMPLE. 
 
 Suppofe the Stock to confift of 20 Cards, 
 and the Ponte's Card four times in it ; 
 y The Banker's Gain ? \\ 
 
 2 X n IX -^3 2x19x17 "" ~~ 'BT*' TT 
 
 nearly. 
 
 Anfiver^ T y T of his Stake accurately, or 
 the eighteenth part of it nearly. 

 
 of PHARAON. 71 
 
 A Table of Pharaon, whereby the federal Ad- 
 vantages of the Banker ', in whatever Cir- 
 
 cumftances he may happen to be^ isfeenfuf- 
 ficiently near by injpetfion, being calculated 
 Jrom the foregoing Theory. 
 
 
 I* Of 
 
 Cards 
 n the 
 Stock. 
 
 The N of times the Ponte's 
 Card is contain 'd in the Stock. 
 
 But 
 
 I 
 
 2 
 
 3 
 
 4 
 
 S^ 
 
 50 
 4 B 
 
 46 
 
 44 
 
 _42_ 
 
 40 
 38 
 
 Jl 
 
 34 
 
 -11 
 
 30 
 
 # * 
 
 18" 
 
 46 
 
 44 
 
 ~94~ 
 90 
 
 86 
 82 
 
 IL 
 
 74 
 70 
 66 
 62 
 
 Ji 
 
 54 
 
 "6T 
 62 
 60 
 57 
 54 
 5 2 
 49 
 46 
 
 L 
 41 
 
 3 
 
 50 
 48 
 _46_ 
 44 
 Jf_ 
 40 
 
 JL 
 36 
 
 34 
 3 2 
 30 
 28 
 
 42 
 40 
 
 3 
 J6_ 
 34 
 
 ji 
 
 3 
 
 28 
 "26" 
 
 28 
 2b 
 
 S^ 
 46 
 
 _^ 
 33 
 
 26 
 24 
 
 i 24 
 
 22 
 
 2O 
 
 ~7cT 
 
 " 16 
 ~M~ 
 
 12 
 
 IO 
 8 
 
 J 
 
 22 
 
 2O 
 
 "T8~ 
 "76" 
 
 14 
 
 42 
 
 _ii 
 
 34 
 3^ 
 
 30 
 
 28 
 
 2S 
 22 
 2O 
 
 J 7 
 
 22 
 
 20 
 
 IF 
 
 16 
 
 Hi 
 
 12 
 10 
 
 T 
 
 26 
 
 22 
 
 ~TT 
 14 
 
 12 
 
 14 
 
 12 
 Q 
 
 1C 
 
 8 
 
 n 
 
 6
 
 72 Of the Game 
 
 But if an abfolute degree of exactnefs be 
 required, it will be eafily obtain'd from the 
 Rules and Examples given at the end of 
 each Cafe. However, to make all things as 
 plain as poffible, I mail, to illuftrate the Ufe 
 of this Table, give an Example or two. 
 
 EXAMPLE i. 
 
 Let it be required to find the Gain of the 
 Banker when there are 30 Cards remaining 
 in the Stock, and the Ponte's Card twice 
 in it. 
 
 In the firft Column feek for the Num- 
 ber anfwering to the number of Cards re- 
 maining in the Stock : over-againft it, and 
 under number 2, which is at the head of 
 the Table, you will find 54, which mews 
 that the Banker's Gain is the fifty-fourth 
 part of his Stake. 
 
 EXAMPLE 2. 
 
 Let it be required to find the Gain of the 
 Banker when there are but 10 Cards re- 
 maining in the Stock, and the Ponte's Card 
 thrice in it. 
 
 Again ft 10, the number of Cards in the 
 firft Column, and under number 3, you 
 will find 12, which denotes that the Ban- 
 ker's Gain in this Circumftance is the twelfth 
 part of his Stake. 
 
 COROL-
 
 
 of PHARAON. 73 
 
 COROLLARY i. From the Con ftruc- 
 tion of the Table it appears, that the fewer 
 Cards there are in the Stock, the greater is 
 the Gain of the Banker. 
 
 C o R o L . 2 . The lead Gain of the Ban- 
 ker under the fame circumftance of Cards 
 remaining in the Stock, is, when the Ponte's 
 Card is but twice in it, the next greater 
 when three times, ftill greater when but 
 once, and the greateft of all when four 
 times. 
 
 Of
 
 74 Of the ACE (/HEARTS ; 
 
 Of the A c E of H E A RT s, or FA i R 
 
 CHANCE. 
 
 THIS Game is pretty much in vogue, 
 as well as that of Pharaon ; there- 
 fore it may not be improper here to touch 
 a little upon the Advantage or Gain that 
 accrues to the Banker or Taliere at this, as 
 well as of that. 
 
 The manner of playing at it is as follows ; 
 There is a Table, on which is painted a 
 felect number of Cards, generally 31 or 25, 
 or between thofe Numbers, limited at the 
 fancy of the Perfon who banks the Table ; 
 the Player flakes upon either of fuch Cards 
 more or lefs, at his pleafure. On the Table 
 is fixed an Engine, called a Worm, into 
 which is put an Ivory Ball, which runs 
 round till it drops or falls into a Socket 
 contiguous to one of thofe Cards; and if 
 it happens to be the Card on which the 
 Player has flaked, he faves his Stake, and 
 is intided to 28 or 23 times as much more, 
 according to the number of Cards painted 
 on the Table, and the cuilom of the Place 
 
 where
 
 dr FAIR CHANCE. 75 
 
 where the Table is kept. From which De- 
 fcription of the Game, the following Pro- 
 blem naturally arifes. 
 
 P R O B. III. 
 
 ffie number of Cards upon fuch a Table ', and 
 the number of Stakes the Ranker pays in 
 cafe he fafes, being given ; to find the Gain 
 of the Banker upon any Siim depofited as 
 a Stake. 
 
 RULE. 
 
 FROM the number of Chances or Cards 
 the Table confifts of, fubtract the num- 
 ber of Stakes more by one than the Banker 
 pays when he lofes, and multiply the Re- 
 mainer by the value of the Stake^ and di- 
 vide the Product by the number of all the 
 Chances or Cards upon the Table, and the 
 Quotient will be the Banker's Gain upon 
 that Stake, and of the fame Denomination 
 with it ; confequently upon two Cards h6 
 will have twice that Advantage, upon three 
 Cards thrice, Gfc . foppofing the Stakes equal. 
 
 EXAM. i. 
 
 Suppofe a Table confifts of 31 Cards, 
 and that the Banker pays 28 Stakes when 
 he lofes ; %uery> His Advantage upon a 
 Stake of 100 / ? 
 
 L 2 31
 
 j6 Of the ACE 0/ HEARTS ; 
 
 3 1 The N of all the Chances. 
 29 The N of Stakes paid more 
 
 by one. 
 
 2 Difference. 
 Multiply by 100 the Stake. 
 
 31)200 (6 /. 
 186 
 
 20 
 
 31)280(91. 
 279 
 
 Anfwer> 61. 
 
 . 
 
 But this Rule is capable of determining 
 the Gain of the Banker when unequal, as 
 well as equal Stakes are fet upon two or 
 more different Cards, it being only to be 
 considered as fo many diftindl Operations. 
 
 EXAM. 2. 
 
 Suppofe A, J9, C play, and flake between 
 them ioo/. viz. A 50, B 30, 20 j Query, 
 The Advantage of the Banker?
 
 or FAIR CHANCE. 
 
 77 
 
 5 
 
 2% 2 Difference, 
 30 
 
 93 
 
 7 
 
 20 
 
 3 1) 140(4 j. 
 124 
 
 ,6 
 
 3 1 
 29 
 
 _ 20 
 
 31)58^(18*. 
 
 3 r 
 
 270 
 
 248 
 
 22 
 
 31) 
 
 2 Diff. 
 
 20 
 "40! I 
 
 20 
 
 Hence theC^l 
 Banker's ^B Ms 
 Gain upon CC J 
 
 Total Gain 
 
 2 5 
 
 The fame as before ; whence it follows, 
 that let the Money be ftak'd how you will, 
 that is, upon as many Cards in what fhape 
 foever you pleafe, the Banker's Gain will 
 be after the rate of 6 /. 9 T ' T per Cent, upon 
 all the Money ftak'd. 
 
 But
 
 78 Of tie ACE of HEARTS; 
 
 But for the fake of a farther Illuftration, 
 let us fuppofe a Perfon to ftake One Pound 
 Sterling upon each Card, to the number of 
 3 1, wz. all the Cards upon the Table ; in 
 fuch inftance, 'tis plain, he's fure of win- 
 ning, or to receive 28 /. for one of the 
 Cards, and fave the Stake of that Card 
 which wins j but then 'tis alfo evident upon 
 the whole, that he mufl be a lofer two 
 Pounds, fincehe wins only 28, and lofes 30. 
 Wherefore by the Rule of Proportion it 
 Follows, that if in flaking 3 1 Pounds he 
 lofes 2, in flaking loo Pounds he will lofe 
 6 /. 9 T f _, which confequently is the Banker's 
 Gain. See the Operation. 
 
 /. /. /. /. s. 
 31: 2 :: 100 : 6. 9T ' T 
 100 
 
 31)200 (67. 
 186 
 
 20 
 
 
 
 31)280(9*. 
 279 
 
 Wherefore
 
 or FAIR CHANCE. 79 
 
 Wherefore the Advantage per Cent, is fo 
 plain, that 'tis needlefs to dwell any longer 
 upon it : However, it may not be amifs to 
 obferve, that 'tis an equal Wager that any 
 one Card will win once in 2 1 times, notwith- 
 ftanding the number of all the Chances arc 
 3 1 i which Event is difcover'd in this or any 
 other Table, by this general 
 
 RUL E. 
 
 From the number of all the Cards upon 
 the Table, dedudt one, and multiply the 
 Remainer by feven tenths, and the Product 
 is the Anfwer. So in this Cafe, 
 
 31 1 =30 
 
 Multiply by ,7 
 
 21,0 
 
 And the Product 21 is the number of 
 Trials requifite for any one Car4 to win 
 upon an equality of Chance. 
 
 Of
 
 So Of LOTTERIES. 
 
 O/*LOTTERIES. 
 
 IN this place I fhall confider the Solution 
 of feveral Problems relating to Lotteries, 
 which may be of ufe to prevent fome mif- 
 takes that People, not vers'd in fuch Com- 
 putations, frequently run into. For in- 
 ftance, in the prefent Lottery for the Year 
 1737, where the proportion of Blanks to 
 the Prizes is as 9 to i, 'tis natural enough 
 to conclude, that 9 Tickets are requifite for 
 the chance of a Prize ; and yet from mathe- 
 matical Principles 'tis evident, that 7 Tickets 
 are more than fufficient for that purpofe, 
 that is, in 7 Tickets it is more likely to 
 have a Prize than not : for this, and all 
 other Cafes of this nature, we {hall give 
 the Arithmetical Solution of the following 
 Problem. 
 
 PROS. IV. 
 
 To find how many Tickets mujl be taken, to 
 make it as probable that one or more Prizes 
 may be taken as not, 
 
 RULE
 
 Of LOTTERIES. 81 
 
 RULE. 
 
 Multiply the number of Blanks there 
 are to one Prize by feven tenths, and the 
 Product is the Anfwer. 
 
 EXAMPL E. 
 
 Query, The number of Tickets requifite 
 in a Lottery, whereof the number of Blanks 
 is to the number of Prizes as 9 to i, to 
 make it an equal Chance for one or more 
 Prizes. 
 
 9 
 7 
 
 , The Product 6,3 mows there is 
 more than an equality of Chance in 7 
 Tickets, but fomething lefs than an equality 
 in 6. 
 
 EXAMPLE 2. 
 
 <%uery, The number of Tickets requifite 
 in a Lottery, whereof the number of Blanks 
 is to the number of Prizes as 5 to i, to 
 make it an equal Chance for one or more 
 Prizes. 
 
 5 
 
 >7 
 
 3*5 
 
 M Anfwer -,
 
 83 Of LOTTERIES. 
 
 Anfwer, The number of Tickets requi- 
 fite to that effect is between 3 and 4. 
 
 PROS. V. 
 
 70 fold bow many 'Tickets miijl be taken, to 
 make it as pr.ebabti that five or more Prizes 
 will be taken as not. 
 
 R U L E. 
 
 Multiply 1,678 always by trip number of 
 Blanks there are to a Prize, and the Product 
 will be the Anfwer. 
 
 EXAMPLE. 
 
 How many Tickets mutt be had in a 
 Lottery, to make it as probable that two or 
 more Prizes will be taken as not, wheri 
 there are 9 Blanks to a Prize ? 
 
 1,678 
 9 
 
 Anfaer, More than 1 5 Tickets, or rather 
 more than 1 6, as mall be prov'd farther on, 
 tho' one might undertake upon an equality 
 of Chance to have one at leaf! in 7 Tickets. 
 
 7'he Numbers 0,7 and 1,678 made ufe 
 of to folve this and the preceding Problem, 
 is the refult of determining the Limits of % 
 in thefe Equations, war. a -\- b x = 2^, 
 
 and
 
 Of LOTTERIES. 83 
 
 and a -f- b* = 2^* -f- iaxb x \ where </ 
 and reprefent the number of Chances re- 
 fpectively for the happening and failing of 
 an Event at any one trial. 
 
 'Tis not my defign to give a Solution to 
 both thefe Equations j but being determin'd 
 to find the Value of x in one of them, I 
 {hall undertake ^he laft, as apprehending 
 it will be mod: acceptable, efpecially to thofe 
 who are not very ready in folving exponen- 
 tial Equations. 
 
 X 
 
 a -\-b === 2b* -j- 2axb*~ l , becomes 
 ^ = 2 -+- 2axlr- 1 by dividing both 
 
 b 
 
 fides of the Equation by b*. Make a : b :: 
 I : 7, and the laft Equaticn by fubftitution 
 
 becomes i -f- ~ = 2 -f- 2f , in which, if 
 
 q be fuppofed equal to J, x will be found 
 equal to 3 5 but if q be fuppofed infinite, 
 
 and = z, x will be infinite alfo, and 
 
 we mail have i -f- = 2x1-1-2?. 
 
 i , 
 
 Again, the Hyperbolic Log. of i -f- 
 
 multiplied by zq is equal to the Hyperbolic 
 Log. of 2 -+- the Hyperbolic Log. df i -\- z. 
 
 But the Hyperbolic Log. of i -t- y multi- 
 plied by == i, therefore z == Hyperbolic 
 M 2 Log,
 
 84 Of LOTT ERIES. 
 
 Log. of 2 4- the Hyperbolic Log. of 
 From which laft Equation z may be found, 
 for z log. i -H z = log. 2 = ,693 147 
 = n 5 but the Hyperbolic Log. of i -f- z 
 
 ZK z.3 z.4 c i r 
 
 == z -- - H ---- , &c. therefore 
 
 x ** * 4 ** e ii 
 
 ~ "~~7 + T~~ T' &c -= ff > and b y 
 
 reverfing the Series z = ^/2n -f- -f- 
 
 to 
 
 1,678 nearly. 
 
 Hence the Value of x in all Cafes will 
 be between ^q and 1,6785' ; but AT converges 
 pretty foon to the laft of thofe Limits, and 
 fo the number 1,678, when x is not too 
 fmall, gives the Anfwer fufficiently exacft ; 
 as in the following Example, where the 
 Odds of the Event's happening is greater 
 than in the former. 
 
 EXAMPLE 2. 
 
 Let it be required to rind in how many 
 Throws, one may undertake upon an equa- 
 lity of Chance, to throw three Aces twice, 
 with three Dice ? 
 
 SOLU TION. 
 
 Out of the 216 Chances upon three Dice, 
 there is but i Chance for three Aces, and 
 215 againfl it ; wherefore multiplying the 
 
 above
 
 Of LOTTERIES. 85 
 
 above Number 1,678 by 215, and the Pro- 
 duct 360,77 (hows, that 360 Throws, or 
 very near it, are requilite to produce the 
 required effect. 
 
 But when x is fmall, as in the preceding 
 Example, it needs a correction j for inflead 
 of 15,102 Tickets, it fhould be 16,443 : 
 which Correction is eafily had by the Rule of 
 double falfe Pofition. For being afTur'd that 
 x is found fomething too little, I therefore 
 affume it equal to 1 6, and fubftitute it in 
 
 the Equation i -f- - = 2 -+- and find 
 
 i i 
 
 the left-hand fide thereof lefs than the right 
 
 by 0,1589 -, wherefore I increafe the value 
 of x four tenths more, viz. to 16,4, and 
 fubftitute it in the Equation as before, and 
 {till find the left-hand fide too little by 
 0,0155; then I multiply crofs-ways, and 
 proceed in the reft of the Operation accor- 
 ding to the nature of the Rule, and find 
 #=16,443. 
 
 However, we are not deflitute of a Me- 
 thod whereby the true value of z, and con- 
 fequently that of x may be found directly, 
 
 by the help of an infinite Series, viz. ~ 
 
 + l? + y^ 3 > &c - For P u ing the 
 
 Hyperbolic Log. of i H = m t mq 
 
 = r, the fhnding Quantity 1,791759
 
 86 Of LOTTERIES. 
 
 = n, and 2.mq n = s. The firft T" errri 
 of the Series, i>/;s. lubtradled from 2, 
 will give the value of z in this cafe, true 
 to two places of Decimals^ viz. 1,83, whence 
 1,83 x 9 = 16,47 ' ls tne va ^ ue f *> or 
 true number of Tickets very near ; for three 
 Terms of the Series make x = 1 6,44300. 
 And if x be fmaller ftill, but fo as not to 
 have the number of Blanks to a Prize lefs 
 than 4,1473 (which feldom or never hap- 
 pens in Lotteries) more places of Decimals 
 will turn out true ; in mort, the above Series 
 will determine the Value of z all Cafes, when 
 q is between 4,1473, and any other num- 
 ber how great foever. 
 
 Though, as it has been obferv'd before, 
 when q. is any thing large, 1,678^ gives 
 x fufficiently near ; for two Terms of this 
 Series, in the cafe of throwing three Aces 
 twice with three Dice, make z === 1,686, 
 and confequently x = 1,686 x 215 = 
 362,49, which is not two throws more 
 than by the former Computation. 
 
 Nofe, The Solution of this Equation, viz. 
 
 a -+- b = zb* -4- 2xab x ~ l -|- x x x i 
 x a'-b* 2 will give the value of x, in the 
 Cafe of a triple Event ; and the Solution o'f 
 
 this, viz. a + b = 2$* -f- 2xab x I -f- x 
 x ^ ^ x ^^a>lf x ^3 
 will
 
 Of LPTTERIE s. 87 
 
 will give the value of x in the Cafe of a 
 quadruple Event. Here follows a Table of 
 the Limits of x from one to fix Events in- 
 clufive. 
 
 The Value of x will always be, 
 
 I 
 < 
 
 ffmgle 
 double 
 
 1 f if] 
 g I 37 
 
 0,693? 
 1,678? 
 
 , 
 quadruple i 
 
 quintuple 
 
 P R o B. VJ. 
 
 The 'Number of Tickets a Perfon has in a 
 Lottery being given, to find the Odds again ft 
 him whether they a// prove Prizes. 
 
 RULE. 
 
 To the number of Blanks to a Prize add 
 T, and make the Sum the Denominator of 
 a Fraction whofe Numerator is Unity ; 
 then multiply this Fraction continually into 
 itfelf as often as the Perfbn has Tickets in 
 the Lottery, bating one, and from the laft 
 Fraction thus produc'd, if Unity be taken 
 from its Denominator, the Remains will 
 fhew how many to I it is, that thsy all 
 prove Prizes. 
 
 EXAMPLE,
 
 88 O/* LOTTERIES. 
 
 EXAMPLE. 
 
 Suppofe I have three Tickets in the Lot- 
 tery of this Year 1737, where there are 9 
 Blanks to a Prize ; how many is it to one 
 but that they are all Prizes r 
 
 9 -4- i = 10 the Denominator. 
 
 v I v I 
 
 To" x To" X TTT Too"o' 
 
 Answer, 999 to i. 
 
 N. B. This Rule is only applicable to 
 Lotteries, or in Schemes where there are a 
 great number of Blanks and Prizes. 
 
 PROS. VII. 
 
 Having the Number of Tickets, and the Num- 
 ber and Amount of all the Prizes undrawn 
 (it any time given, to find the Value of a 
 Horfefor any number of Days. 
 
 RULE. 
 
 Multiply the number of Prizes by the 
 Price of an undrawn Ticker, and fubtract 
 the Product from the Amount of all the 
 Prizes, and multiply the Remainer by the 
 number of Days the Horfe is hired for, and 
 referve the Product for a Dividend ; then 
 multiply the number of undrawn Tickets 
 by the number of Days required to draw 
 them in, and with the Product divide the 
 aforefaid Dividend, and the Quotient will 
 s be the Value of the Horfe. 
 
 EXAMPLE.
 
 Of LOTTERIES. 89 
 
 EXAMPLE. 
 
 Let it be required to find the Value of a. 
 Horfe for the firft Day's drawing in the" 
 prefent Lottery, where there are 70000 
 Tickets at lo/ each, number of the Prizes 
 7000, amounting to 22^000 /. exclufive of 
 the two Prizes for the firft and lafl Num- 
 bers drawn, viz. 500 and iooo/. arid let 
 us fuppofe that the whole time of Drawing 
 will be 40 Days. '.U 
 The N of Prizes ^ -- ----- 
 
 Price of an undrawn Ticket - - 10 
 
 70000 
 
 The N of undrawn Tickets 70000 
 
 The whole time of drawing - 4.0 d. 
 
 2600000 
 
 The Amount of all the Prizes; deduc- 
 ting 14 per Cent, is 1943607. 
 194360 
 
 70000 
 2800000) 124360,000 ( ,044 
 
 I 1200000 20 
 
 L 5 88o 
 
 12360000 
 
 "30000 
 
 Il6oOCO r 
 
 /2 3 2 4 
 
 Anfaer, i o d. |. 
 
 N Thofe
 
 go Of LOTTERIES. 
 
 Thofe that do not chufe to divide deci- 
 mally, may, if they pleafe, multiply the 
 aforefaid Dividend by 960 before they di- 
 vide, and the Quotient will be the Anfwer 
 in Farthings -, e . g. 
 
 124360 
 960 
 
 7461600 
 1119240 
 
 28)00000) 1 193(85600 
 ~42~~ ""112 
 
 4) 73 
 
 101 56. 
 17 
 
 Which i o A. 4- would be the real Value 
 of the Horfe j but as it's on the firft Day's 
 Drawing, there is a probability of its being 
 the firft drawn, in confequence of which 
 the Owner is intitled to a Prize of 500 /. this 
 Expectation is worth about three Half-Pence 
 more, 'viz. the 70,000 part of the Value of 
 500 /. when the 14 per Cent, is taken off, 
 which being added to the Value before 
 found, makes i Shilling, the mathematical 
 Value of a Horfe for the firft Day's drawing. 
 
 EXAMPLE 2. 
 
 Admit that 1 2 Days before the end of 
 the Drawing, there are left in the Wheel 
 
 of
 
 Of LOTTERIES. 91 
 
 of Fortune 20544 Tickets, of which 2100 
 are Prizes, amounting in all to 714007. 
 Query ', The Value of a Horfe for two Days, 
 the Price of an undrawn Ticket at that 
 time being worth 1 1 Pounds. 
 
 The Amount of all the Prizes, deduc- 
 ting 1 4 per Cent, is 61404 Pounds. 
 
 The Number of Prizes . . 2 1 oo 
 
 The Price of an undrawn Ticket 1 1 
 
 2100 
 2100 
 23100 
 The Number of undrawn Tickets 20544 
 
 Number of Days to draw them in i_z 
 
 246528 
 61404 
 23100 
 
 246528)76608,000 ( ,310 
 
 739584 _20_ 
 
 1 S. 6,200 
 
 264960 12 
 
 246528 
 
 l8 43 20 
 Anfwer> 6 s, ^d. . 
 
 N 2 Note,
 
 ga Of LOTTERIES. 
 
 Note, Irrthefe Calculations' 'tis fuppofed^ 
 (as is cuftomary) that if the Horfe proves 
 a Prize during your Jockeyfhip, that i o /. 
 or an undrawn Ticket be rtftur'd to the 
 perfon who let it. 
 
 It is alfo very plain that this Rule will 
 ferve to value the Chance of a Ticket du- 
 ring the whole Lottery, aJlume the number 
 of Days it will take in drawing what ycu 
 pleale -, wherefore to render' the Opera uon. 
 ea'fy, fuppofe i Day. ' Ar.d altho* I have 
 made a deduction of \^per Cent, upon the 
 Amount of all the Prizes in the two pre- 
 ceding Exr.mples, and at the fame time 
 allow the Calculation to ue Agreeable to the 
 kules of Art and Science, as being founded 
 on the ftricttft Demonflration j yet in the 
 preferit Cr.f ..-, I lay, the Dedudion is not 
 quite realbnable between Buyer and Seller, 
 and confequently none fhould be made in 
 finding the value of a Chance for the whole 
 time : Foi how can irbe expected that any 
 one will give up his right in a Ticket if it 
 proves a Prize, and fbnd to the lofs of a 
 Guinea extraordinary' if it proves a Blank, 
 uiz. the Djfcount upon 7 /. i o s. at the rate 
 of i^per Cent, without a valuable coniide- 
 ration, which is that of taking Chances, 
 valued according to the full Amount of the 
 Prizes, and the Price the Tickets bore when 
 they were firft purchas'd, viz. io/. each.
 
 Of LOTTERIES. 97 
 
 This Method of proceeding will put both 
 Parties upon an equal footing, than which, 
 I think nothing can be more fair and equi- 
 table. 
 
 Hence I make the Chance of a Ticket 
 for the whole time of drawing to be worth 
 2 /. 4*. 6J. , which, with 2d. f for the 
 Expectation of its being either the firft or 
 laft drawn, makes 2/. 4*. gd. f ; but if 
 the Chance happens to prove a Prize, io/. 
 more, or the Price of an undrawn Ticket 
 muft be advanc'd. -However, the Market- 
 price determines what muft be given after 
 all ; wherefore if a Chance mould fell for 
 more than what this Calculation makes, it 
 is not to be wonder 'd at, fince fome con- 
 iideration ought to be made for the rifque 
 that the Dealers in Tickets run in having 
 them fold under par, and for fome con- 
 tingent Expences they are unavoidably at, 
 in furnifhing thofe with Chances and Tickets 
 who are willing to be in Fortune's Way. 
 
 But as in all Lotteries Succefs is preca- 
 rious, we being kept in fufpence till the 
 Event makes known either our good or 
 bad Fortune, fo from luch ftate of Uncer- 
 tainty it follows, that before the Drawing 
 is finim'd a Ticket may be fold for more 
 or lefs than at prefent ; I mall therefore, 
 before I conclude this Subject, mew how 
 its real worth may be known in any cir- 
 2 cum fiance
 
 94 Of LOTTERIES. 
 
 cumftance of the Lottery, by which means 
 the value of a Chance may be very accu- 
 rately determin'd at the fame time, e. g. 
 
 Multiply the Number of Blanks remain- 
 ing in the Lottery at any time of its draw- 
 ing, by the Price of a Blank, which is al- 
 ways fix'd, and to the Prbduct add the 
 Amount of all the Prizes remaining, the 
 laft drawn included; this Sum divided by 
 the Number of all the Tickets, izz. Blanks 
 and Prizes, will give the value of an undrawn 
 Ticket, which being known, the Value of 
 a Chance for the time, during the remaining 
 part of the Lottery, eafily flows from the 
 afore- mentioned Rule, 
 
 I mall conclude this fmall TracT: by ma- 
 king fome Remarks, relating to Hazard and 
 Backgammon j the truth of all which is 
 cafily deduc'd from the preceding 
 tions*
 
 Of HAZARD. 95 
 
 Of HAZARD. 
 
 i. TF 8 and 6 are Main and Chance, one 
 J[ may lay 155 to 169, or u to 12, 
 that either one or the other is thrown off 
 in two Throws. 
 
 2. And if 5 and 7, or 9 and 7 are Main 
 and Chance, the probability of their being 
 thrown off in two Throws is alfo as 155 
 to 169, or as u to 12. 
 
 3. If 5 and 8, or 9 and 8, or 5 and 6, 
 or 9 and 6 are Main and Chance, the pro- 
 bability of throwing one of them off in two 
 Throws is as 7 to 9 exactly. 
 
 4. And if 7 and 4, or 7 and 10 are 
 Main and Chance, the probability of their 
 being thrown off in two Throws is alfo as 
 7 to 9. 
 
 5. If 7 and 8, or 7 and 6 are Main and 
 Chance, one may lay 671 to 625, or 15 
 to 14 that one of them is thrown off in 
 two Throws, fo he that lays an even Wager 
 he will throw one of them off in two 
 Throws has the beft of the Lay.
 
 Of HAZARD. 
 
 6. But if 5 and 4, or 5 and 10, or 9 
 and 4, or 9 and 10 are Main and Chance, 
 he that undertakes to throw either Main or 
 Chance in three Throws has the worft of 
 the Lay ; for it is as 22267 to 24389, or 
 in fmaller Terms, as 21 to 23 exceeding 
 near ; the Ratio of 21 to 23 differing from 
 that of 22267 to 2 43^9 only but by the 
 ten thoufandth part of an Unit. 
 
 Note alfo, that n and 12 exprefs the 
 Ratio of 155 to 169 the neareft poffible in 
 fuch fmall Terms, as does 15 to 14 that of 
 671 to 625, and are eafily difcover'd by 
 the Rule exhibited in my Appendix to 
 Dr. Ketl's Euclid. 
 
 7. Suppofe IV to be a Main, and the 
 Law of the Hazard to be this ; That if the 
 Cafter throws either II, III, IV, XI, or 
 XII the fii-ft Throw, he mall lofe the whole 
 Stake, and if he throws V, VI, VII, VIII, 
 IX, or X, either of which, as it may hap- 
 pen, mall be deem'd a Chance againfr. IV, 
 fo which ever comes up firft wins ; Query % 
 The Hazard of the Cafter to that of the 
 Setter ? 
 
 Anjwer^ The Hazard of the Carter is to 
 that of the Setter as 457 to 551, or as 5 
 to 6 very near ; wherefore the Gain of the 
 Setter, each Stake being a Guinea, will be 
 T y_. equal to i s. 1 1 d. ? exactly. 
 
 8.
 
 Of HAZARD. 97 
 
 8. And at Hazard, if the Main be y t 
 and each ftake a Guinea, the Gain of the 
 Setter is about 3^. 
 
 9. If the Main be 6 or 8, the Gain of 
 the Setter is about Six-pence in a Guinea. 
 
 10. But if the Main be 5 or 9, the Gain 
 of the Setter is about 3 d. -J- in a Guinea ; 
 whence it follows, that 5, 7 and 9 are 
 much upon a par to fet on, and that 6 and 
 8 are fomething more advantageous. 
 
 11. However, if a Perfon is determined 
 to fet upon the firfl Main that is thrown, 
 his Advantage, fuppoling each Stake to be 
 a Guinea, is the T |-rr f a Guinea, which 
 when reduc'd will be found equal to 4^. |, 
 and half a Farthing exactly. 
 
 12. Hence the probability of a Main, to 
 the probability of no Main ; or, to fpeak 
 in trie gaining Phrafe, a Main or no Main, 
 is as 2016 37 to 20 1 6 -f- 37 ; that is, 
 as 1979 to 2053 accurately, or as 27 to 
 28 very near ; for if one flakes 27 Guineas, 
 the other ought not to flake quite 2 d. -*- 
 more than 28 Guineas, which is a fmall dif- 
 ference from the truth in fuch large Stakes 
 as 27 and 28 Guineas. 
 
 13. If, with two Dice, one mould under- 
 take to throw firft the two Aces, next the two 
 Duces, next the two Threes, next the two 
 Fours, next the two Fives, and laflly the 
 two Sixes, the Odds again ft him would be 
 
 O
 
 g8 Of HAZARD, 
 
 two thoufand, one hundred, and feventy- 
 fix Millions, feven hundred, eighty-two 
 thoufand, three hundred, and thirty-five 
 to one ; and tho' this might poffibly hap- 
 pen the firft fix throws, yet the Odds are 
 fo immenfely great againft it, that it Would 
 probably require whole Ages to perform 
 it in : yet notwithstanding all this difficulty 
 in throwing firft the two Aces, next the 
 two Duces, &c. they may with an equa- 
 lity of Chance be undertaken to be thrown 
 in lefs than a quarter of an Hour, in the 
 following manner, viz. to throw away till 
 the two Aces come up, then till the two 
 Duces, then till the two Threes, and fo on 
 till the two Sixes are thrown ; but to throw 
 them fucceffively is what, never yet, is ra- 
 tional to fuppofe,fcas been done by any one. 
 14. If any one mould undertake to throw 
 a Six or an Ace with two Dice in one 
 throw, he ought to lay 5 to 4, whereas 
 'tis ufual to lay an even Wager only; in 
 which cir cum fiance the Caller has fo much 
 the better of the lay, as in the long run to 
 impoverifh the beft Eftates, not to fay ruin 
 them. Tho' at firft fight it muft appear 
 to an Eye not vers'd in thefe Speculations, 
 a little odd, that the Setter mould not have 
 the beft of it, fince there are but two Sixe?, 
 and two Aces for the Cafter, and two Fives, 
 ,two Fours, two Threes, *and two Ducesj 
 
 for
 
 Of .BACKGAMMON. 99 
 
 for the Setter. And were the Points > of 
 both Dice all made upon a regular Solid or > 
 Body of 1 2 equal Faces, fuch as the Dode-' 
 caedron, the Cafter would undoubtedly have 
 the advantage ; for then he would have two 
 to one of the lay, in as much as he would 
 have 8 Chances for winning, 'and but 4 foV 
 lofing : But as there are two Dice, it mufl 
 be confidered as the happening of two 
 Events, independent of each other, which 
 makes the Odds, as I faid before, jufl 5 to 
 4- 
 
 Co^OLL AR Y. 
 
 Hence it follows, that 1 at 'Backgammon 
 if two Points are open, 'tis 5 to 4 but that 
 a Perfon enters the firft throw ; and as this 
 Thought naturally leads me to give a Solu- 
 tion of the reft of the Hazards, it may per- 
 haps be acceptable if I mew the Odds of en- 
 tring when other Points of the Table are 
 open, and therefore mall give the following 
 Scheme for that purpofe. 
 
 SCHEME. 
 
 Points open 
 
 A Perfon may lay 
 
 i 
 
 III' f2cf 
 
 2 
 
 3 
 
 jlj 
 
 i > that one enters. 
 
 4 
 . jf 
 
 35J 
 
 j] 
 
 And
 
 ioo Of BACKGAMMON. 
 
 And I don't doubt, but the Knowledge 
 of thefe Odds may enable one to play the 
 Game in other refpeds with great advan- 
 tage j tho* for my part I own with regard 
 to pra&ice, that I have but very little ikili 
 in this, or any other Game whatfoever. 
 
 FINIS.
 
 A 000 031 305 6