Columbia Mmtiersitp intljeCup ofJtrtoPork THE LIBRARIES THE SELIGMAN LIBRARY OF ECONOMICS PURCHASED BY THE UNIVERSITY 1942 H1A&_Ay 33- A LETTER T O T H E GAMESTER. Numb. I. WHEREIN The Miftakes of that Author’s Calculation, and the Unreafonablenefs of his heavy Cenfares are plainly demonftrated. WITH Some Advice to the STOCK-JOBBER others concern’d in Lottery-Tickets , in Relation to the Two different Sets of Propofols for Injuring Ticket! ’ at Mercers-Hall. 4 • - Habeat jam Roma pudorem. Tertiui eCcelo cecidit Cato. Quid dignum tanto feret hie Promijffbr hiatu ? Parturiunt Montes. Juv. Hor. LONDON; Printed for T. Bicker ton, at the Crown inTater* noffer-RoWy 1719. Price 4 4. A T O T H E GAMESTER. Numb. I. SIR, INDING you engaged in fo ufeful a Defign, as than of preventing the Publick from being impofed upon by the Deceits, that are lo fre- quent in Gaming, I amrefolv’d to be yourCor- reipondent. And as I am fenfible of the Difficulty of the Task you have undertaken, as well as of your want of Capacity for performing it, I think I can no way be more ufeful to you and your Readers, than by laying open from Time to Time, luch Miftakes, as I perceive you will unavoidably fall into, not in the lead doubting of a Re- A turn turn of the fame good Office, whenever I ffiall have occa- fion for it. Therefore Ceremonies fet afide, as among thole who refolve to be Friends, I fhali in few Words acquaint you with the Reafon of entring into this Cor- refpondence. An Acquaintance of mine, who is one of thofe ufeful Perlons, vulgarly call’d Stock-Jobbers (for fuch I ffiall always efteem thole Gentlemen, who, by affording a ready Market at all Times for Government Securities, do thereby preferve their Credit, and keep up their Value,) happen’d to purchale a great Number of Tickets in the firft of the prefent Lotteries, at a conffderabl e ( Prcernium. Thele, you know, immediately funk in their Value, upon voting the Second Lottery, and my Friend not eaflly refolving to ffiffer lo great a Lofs, as he mull have done by dilpo- fing of his Tickets at the Market-Price, was determin’d to keep them till the Drawing, at which Time there was a general Expectation they would rile. In this he was again difappointed, the Tickets being confiderably under iVar , both before and after they began to be Drawn. Thefe Circumllances very much perplex’d him, as having already loll the entire ‘Prceminm , which he had paid for his Tick¬ ets, and being now reduc’d to the Neceffity either of felling his Tickets at about 8 per Cent, below ‘Par, or elfe of running the Rifque of the Wheel. The firft he could by no Means refolve upon, and to do the latter might very much incommode him in his Fortune, in cafe of ill Succefs. In this Exigence he turn’d his Thoughts upon infuring with the Company at Mercer's-Hall , by which Means he propofed to himfelf to take the Advantage of what good Fortune might happen, and at the fame Time not to run the hazard of a greater Lofs, than what he knew, he was able to bear. His Reafon indeed told him, that the Gentle¬ men, who had lublcrib’d lb great a Sum as iaoooo /. had no doubt confider’d very well of their Project, and had it examin’d by good Hands, before they engag’d in it, and therefore in all Probability mud make a reaionable Gain out of the Tickets that were infur’d with them. But this he was willing to comply with, for the fame Reafons, that ufually determine the more prudent Adventurers in Ships and Merchandize to infure their Effebts. In this Juncture came out your Performance, which my Friend, with many others, was allur’d by the fpecious Ti¬ tle to purchafe and read over. But finding by your Com¬ putation, that the Company were like to gain above 5300c/. out of each Lottery, in cafe they inlur’d the whole, which is more than 10 per Cent, he now found himfelf again at a Lois, and inclin’d rather to fell his Tickets at 7 or 8 per Cent, difcounr, than to allow thole Gentle¬ men lo unreafonable a Profit. While he was under thele Doubts, I happen’d into his Company, and as he had never made any Scruple of ac¬ quainting me with the State of his Affairs, he unbofom’d himlelf to me upon this Occafion, and defir’d my Advice, Ihewing me at the lame Time your Computation. I had not at that Time confider’d the Matter, but however, upon running over your Scheme, I prelently dilcover’d le- veral Miftakes iti the Calculation, by which I convinc’d him, that you had carry’d the Advantage of the Company much too high, and promis’d him, that, if he would ftay two or three Days, I would examine that Affair more particularly, and would acquaint him with the Refult. When I had done this, I found the Company were fo far from making making above io per Cent, that they could not propole to gain more than 44 per Cent, upon their firft Propofals, and not quite 34 upon their Second, whereby the Adventurers were allow’d to keep their Tickets in their own Hands. Up¬ on which my Friend, who, though he was not qualify’d to go through with the Computation himleif, yet repos’d a good deal of Confidence in my Accuracy and Diligence, was immediately made eafy, and infur’d his Tickets with the Company upon their laft Propolals, whereby he pays 1 6 l. xos. Inlurance-Money for every Set of 12 Tickets. Now, Sir, as this is probably the Cafe of many other Perlons, befides my Friend, I hope it may not be uufer- viceable to the Publick, to prefent them with my Compu¬ tation, by which I hope to oblige them and you at the lame Time. But firft of all it will not be improper to de¬ ceit the Miftakes of that Calculation, upon the Authority of which you have had the Affurance to treat a Set of Gen¬ tlemen of Subftance and Reputation, like fo many Sharpers giving their daily Attendance in Committee, to take in ig¬ norant People upon fuch Lays, as you reprelent to be ex¬ ceedingly fraudulent and difadvantagious. But before I do this, in order to lliew my good Nature and Difpofition to agree with you as far as poffible, I lhall allow you to take it for granted, thatinluring Sets of iz, 24, 48, or 96 Tickets, at different Prices, is equally ad¬ vantageous to the Company. Though, by the by, I can venture to allure you, that they make a greater Adantage by infuring the higher Sets, than an equal Number of Tickets in the lefler Parcels. But as I have neither Leifure ror Inclination to go through with thole great Computa¬ tions, I cannot acquaint you what is the exa£t Diffe¬ rence, nor is it at all material to our prefent Purpofe. I I {hall therefore proceed to examine your Calculation of the Advantage made by the Company in inluring the Sets of n Tickets. Every one of thefe, you fay, is entitled to two Benefits, and therefore, you fuppofe, that they Jhall happen to be drawn exactly two Benefits in a Set , upon which Suppofition your whole Computation entirely depends. By my Calculation, the whole Number of Chances upon n Tickets, for drawing either all Blanks, or any Number of Prizes not exceeding 12, is nearly 1177,000000. Out of thefe the Number of Chances for 7 drawing all Blanks is S x 44 > 000000 For drawing only one Prize ' 586,000000 Only two Prizes 645,000000 Three Prizes, or more 701,000000 From which it appears, that 1 have almofl; as great a Number of Chances for drawing only one Prize in iz Tick¬ ets, as for drawing only two; and that I have a greater Number of Chances for three Prizes or upwards, than for two only Confequently you might more probably have fuppos’d, that a Proprietor of 11 Tickets Ihould have had three Prizes or more, than only two. By which the Rea¬ der will eafily judge, what Strefs is to be laid upon a Com¬ putation, that has no better Support, than luch a Suppofition, Having laid this Foundation, for want of a better, y ou proceed to tell us, There are 1711 Sets , that will happen with one Brize of 10 1, and one of 15, or a greater in each Set, and there will be 11334 Sets, that will happen with two 10 1 . Benefits in each Set. B This This I fee no manner of Reafon for ; for admitting that there fhall always happen two Benefits in each Set of ix Tickets, yet certainly it will fometimes fall out, that two Prizes of xy l. or upwards, fhall arife in the fame Set. From which it follows, that there will be fewer than 17x1 Sets, that fhall happen with a Prize of 10 l. and a greater, and more than 11334 Sets that fhall have two Benefits of 10 l. each ; and as often as this happens, the Infurers mull: pav to the Adventurer 1 61 . infteadof 1 /. in order to make up his Sum of 3 61 . This you feem in fome meafure fenfible of in the third Article of your Computation, where you confider what the Infurers will have to pay to the Adventurers upon thofe Sets of Tickets, in which a xy/. Prize fhall come up with one of 10/. For there you tell us, that fuppofe all the xyl. Prizes fall out fo> (which is very improbable , for fome may and will happen with a greater ‘Prize) there can btit 1408 happen fo , which , as the Company in this cafe is to pay to the Adventurers but 1 /. upon each Set, will amount to but 1408 l. This indeed is very true, but then from the very Reafon that you here give, why this Article can rife no higher than 1408 /. it follows that the next ought to be confider- ably enlarged, fincefor every xor. that you take from this Head on account of axy l. Benefit coming up with a greater Prize, you ought to add 1 61 . to the next, as has been al¬ ready fnewn. From this it appears, that even upon your own Suppo- fition of two Prizes arifing in every Set of ix Tickets, a confiderable Deduction muft be made from the y3ooo/. which which you allow the Company for Profit. And if we re¬ ject this Suppofition as arbitrary and unreafonable, it will lometimes fall out, that four or five of the fmall Prizes lhall arife in the fame Set, as likewife that one of the greater Prizes lhall fometimes fall in with one or two more of the fame, or at lead with two or three of the fmaller ones. In all which Cales the Fortunate will be Gainers, and the Company mult makegood the Deficiency to thofe Adventurers who want their Proportion. You own indeed, to do you Juftice, that it is not Ma¬ thematically pojfible for the Benefit Tickets to arifie fo jufily, as you had before luppos’d; but then you tell us, that what Benefits fall Jhort in one Set , will exceed in ano¬ ther. Which Reafon is fo far from doing you any Ser¬ vice, that it is the very Foundation of my Argument againft you. For where the Benefits exceed in any Set, the Ad¬ vantage is generally the Proprietors, and the Company is oblig’d to makegood the Lois, where-ever they falllhort. You come now to confider the Matter more exactly, and to make it as plain as poffible to every common ‘Vn- derfianding , to prevent ‘People's being gull'a of their Mo¬ ney , and becoming Bubbles , as the ufual Term is amongft Gamefiers. In order to which, you premife the Company’s fecond Set of Propofals, whereby the Adventurers are to pay 1 6 /. ioj. Infurance-Money for every Set of iz Tick¬ ets, to allow the Company no Pramium out of the Prizes, and ro keep their Tickets in their own Hands. Your Ob- fervations upon thefe are lb extraordinary, and fo far from being plain to my Underltanding, that, for the fake of thofe Readers who may have more Penetration than my felf, I lhall give them here in your own Words. [ IO ] They tell you. That to accommodate fuch Perfons, who choofe to keep their Tickets in their own Hands, and pay no Praemium on their Prizes, that is, to allow the 5 per Cent, you may pay 27s 6 d. prefent Money each Ticket, that is, 2 s. 6 d. per Ticket, in lieu of the 5- per Cent, on the Prizes : So that then your Infurance of 12 Tickets will coft you 161. 10 s. for which they agree to pay you 36 1 . if no Prizes, or if Prizes, to make them ups 61 . that is as much as tofay, they will lay you 19 1, 10 s .to 1 61 . 10 s. which is 13 to 11, that the Adventurer will have one or more Benefits in his Set; if he has none, then they lofe their 19 1. 10 s. for with the 161 . 10 s. they are to pay them for their Infurance, and their 19 1 . 10 s. makes up the 36 1. they are to pay : But if he has any Benefits, they are to have them, to help 'em to make up their 19 I. 10 s. So they lay the Adventurer but 13 to n, and take his Benefits, as far as is fufficient, to make up the 19 1. 10 s. Whereas, the Bett is, as may eafily be demon ft ra¬ ted, 1932,641711 to 244,140(525’ which is above 7 to 1, that any Adventurer has one or more Benefit Tickets in 12 Tickets. How is it pofftble there fioould be fuch Fools to bett their Money with thefe Gentlemen at fuch Difad- v ant age ? For if they had a mind to lay more Money upon the Lottery', they had better go to White\r, or the Groom- Porter \r, where they will find Gentlemen enough will en¬ tertain 'em upon a more moderate Lay ; nay, they’ll be glad to lay 13 to 9, that they have two or more Benefits iit 12 Tickets, and the Adventurer to have the Benefit of his own Prizes, which is much better than 13 to 11, This latter part of your Advice, I confefs, is very plain, and very kind to us Citizens, and no doubt every prudent Man among us, that is inclin’d to Gaming, will think him- 1 felf ( ” ) felf very much oblig’d to you for recommending him to Places, where he may meet with luch civil Ufage. For if the Gentlemen at White's, and the Groom-Torter’s , will take him up, upon his laying 9 to 13, when the Bert is a little more than 8 to 13, it’s plain, that upon equal For¬ tune, and fair Play, he can’t loie full 7 per Cent, of his Money upon every Stake ; whereas this Jharping Commit¬ tee of 'Directors are fo unconfcionable as to lay him no more than 13 to 11, according to your Computation, up¬ on a Bett, where the Odds are above 77 to ii ; by which he mull lofe above 31 per Cent, of his Money, as may eafily be demonftrated. Which makes it very natural in you to wonder, that there Jloould be Juch Fools , as to bett their Money with them upon fuch Difadvantage. But my great Difficulty is, that I am perfe&ly ar a lofs, how to Bate the Wager between the Company and the Ad¬ venturers; and therefore, dear Sir, thenext Time you ap¬ pear in Print, be fo kind as to inform me, and the reft of your gentle Readers, how I can lay another Perfon 13 to 11 upon a Wager, and yet lofe my Money upon both fides of the Queftion, which, as you have ftated it, feems to be the Company’s Cafe. To make this Matter as plain as poffible, I beg leave to put a Cafe or two. I infure iz Tickers with the Compa- panyat 1 61 . ioj. upon their laft Propolals. If I have no Benefit, they pay me 36/. that is, they lofe 19 l 10 s. If I draw a great Prize, or a Number of imall ones to the value of above 361 . they get my 16I. 10 s. and fo far the Thing looks like a Wager. But if this were what you meant, you Ihould have laid the Company would lay 13 to 11, that the Adventurer Ihould have one or more Benefits C in ( 12 ) in his Set of iz Tickets to the value of above 3 61 . againd his having no Prize at all, which would make a very diffe¬ rent Wager from that you have flared. But let us fuppofc, that I get one Prize of 10 /. only in my Set of iz Tickets. Mufl not the Company then pay me x 61 . ? If they do this, it’s plain they lofe 9 l. 10 s. So that, whether I get no Prize, or only one of 10 /. they mufl be Lofers. If 1 get a Prize of 25- /. they gain indeed, but ’tis only 5 /. 10 s. inflead of 1 6 1. 10 s. which they ought to gain by your Wager. If I get two Benefits of 10 /. each, they gain 10 s. If one of 10 /. and one of 2.5 l- they g ain *5 k los - if three of 10 /. each, they get 10 /. 10 s. So that, of all the feveral Cafes we have put, there is only one for the Company’s lofing 19/. 10 s. and only one for their gaining 1 61 . 10 s. In all the other Cafes they gain or lofe different Sums, aud confequently you have dated the Cafe in the mod ablurd and confus’d manner imaginable, which you had undertaken to make as plain as pojfible to every common ‘Vnderfianding . I pafs over your Calculation of what the Chances of the Benefit Tickets are worth taken feparately, againd which I have nothing to objedt. Nor have I much to fay about the Inferences you draw from it, as not feeing how far they can be ufefuland beneficial to the Proprietors, or the con¬ trary. I mud only take notice to you, that, when you fay, the fimall ‘Prizes will be of no Advantage to the Proprietor, in cafe he injures , and that, if he has not a larger ‘Prize, he had better have none , you are guilty of a great Midake. I allow you, that, if his Share of thofe Prizes amount to lefs than 36 l. he may pofiibly be fomething a lofer by theDifcount upon the Prizes; but I beg you will confider ( 13 ) confider likewife, thar, if he gets fo many of them, as to make up more than 36 /. againft which there is no great Odds, he may be a confiderable Gainer. I hope, Sir, by this Time, I have convinced you, or your Readers at leaft, that you are no way qualify’d for thofe great Undertakings you pretend to, of paying the ‘Debts of the Nation, and thereby excelling the Famous Mr. Laws hhnfelf of determining the Chances of Cities and Countries taken , Fleets and Armies destroy'd, or of' Deace andJVar . Which Jail appear to me to depend up¬ on 10 many Caufes not reducible to any Calculus , that, I confefs, I ihould defpair of feeing them brought to any Certainty, though De Moivre himfelf were to undertake the Computation. I come now to prefent you with my own Calculation of the Advantage upon infuring Lottery-Tickets, and hum¬ bly hope, that if you difcover any Error in if, you will afford me your charitable Affiftance to fet it to rights, as- you lee I have done by yours. I have 244,140625" Chances our of 2176,78x336 for getting no Prize at all in a Set of ix Tickets, lb that the Probability of my getting no Prize may be exprefs’d by the Vulgar Fra&ion Or by the Decimal Fraction, 0,11211 If this happens, I lhall receive from X / $ the Company -> ’ ~ J So- ( H ) So that the Value of my Expedition from the Company upon thefe Chances, is equal to the Product of the Sum expected, namely 3 6 l. mukiply’d by the probabi¬ lity’ or the Fradion 0,11211, that is to The Probability of my having One finale Prize of 10 /. is the Decimal Fradion° The Sum to be paid me in this Cafe by the Company is 3 And the Value of my Expedation is the Produd of thefe two, or The Probability of one Prize of 2f /. is Sum expected Value Probability of 2 Prizes of 10 /. each Sum expeded Value Probability of one Prize of 10 l. and one of 15- /. Sum expected Value Probability of 3 Prizes of 10 /. each Sum expected Value Value of my Expedation upon no Prize Upon one Prize of xo l. One of if /. Two of 10 /. each One of 10/. and one of 2 c /. Three of 10 /. each l . 26, l. 6 , l. II, l. o, /. 16. l - 3 , l. 1 , /. o, /. 6 , /.o, * 4 , Value of all the Company my Expedations from /.if, dec. 036 o, 24312 dec. 3211 o, 01348 dec . 1483 o, 241 f4 dec. 8646 o, 02679 dec. 0268 °,' 14744 dec. 8726 dec. 0360 6 , 32-11 o, 1483 3 ) 8646 o, 0268 o, 8726 dec. 2694 Paid C 15 ] Paid to the Company, upon their firfl: Propofals, for inluring n Tickets, at t /• ^ €16 C * O Allow’d them upon my Share of the Prizes, at 5 per Cent. /. 01, dec. 8 /. 16, dec, 8 /. if, dec. 1694 Received in all by the Company DeduCt ___ Company’s Profit upon n Tickets l. 1, dec. 5306 From which it follows, that their Pro¬ fit, in Cafe theyinfure a whole Lottery, will be about „ . nJ , /lIJ00 ’ Which is nearly 47 per Cent, mltead of 10 per Cent, or /. 53000, by your Computation. By their fecond Propolals, the Company receive for inluring 12. Tickets at 2,7 s. 6d. each. />l6 ’ deC ‘ S From which deducting the Value of the Adventurers Expectations. /. 15-, aec.% 694 The Company’s Profit upon 12. Tickets is /. oi, dec. z$o6 So that their Profit in Infuring a whole Lottery upon thefe Propofals would be a- bout Which is lefs than 3^ per Cent, and will I believe be thought no unrealonable Profit, confidering their Ciiafge^ [ . 1 * ] and the conftant Attendance of lb many Gentlemen, whole Time cannot but be very valuable. From this Computation it follows, that to gain izoo /. the Company mull infnre 9400 Tickets upon their firft Propofals, or 11700 upon their fecond. And to gain 3000/. they mult inline z3 joe) Tickets up¬ on the firft, and Z9Z50 upon their laft Propofals. Which I mention, not upon your Account, but for the fake of that publick-fpirited Perfon, the Projector of this Info* rance, who no doubt underltands the Meaning of this Cal¬ culation. Having now gone through with what I propos’d, I come to confider what Service I can do my Reader, in order to enable him fome way or other to make up the Six pence, which my Pamphlet has coll him. And accordingly, in the firft Place, in cafe he is inclin’d to inlure his Tickets at Mercers-Hall, I advife him to put in his Tickets in Sets of iz Tickets each, the larger Sets being more to his Difadvantage, than the fmaller. In the fecond Place I caution him to inlure upon the Company’s fall Propofals, thofe being more advantageous to him than the firft, by above f per Cent. Which Advice, I hope, will convince all the World, that I am not brib’d by the Company to this Undertaking. Aud therefore I charge you, as you value the good Opinion of your Readers, not to drop any Infinuation of that kind, in cafe you favour me with an Anfwer. Laftly, N C 17 ] Laftly, to gratify thofe of my Readers, who may be inclin'd to Bett Money upon their Tickets, I here prefenc them with an Account of the exadt Odds to be laid againft zos. that one or two Benefits lhall be drawn in any Num¬ ber of Tickets not exceeding iz. Tickets. i- z- 3 - 4 ' 5- 6 - 7- 8 - 9“ 10- 11- iz~ Odds for i Prize. /. s. d. q. -oo 04 00 o— -00 08 09 1— —00 14 06 3— —01 01 05 3— —01 09 09 I— —01 19 08 3— —oz II 08 o— —03 06 00 o— —04 03 oz z— -05 03 10 o— -06 08 07 I- •o7 18 03 3- Odds for z Prizes. /. -01 s. d. f- 00 CO 0 00 06 3 01 07 1 03 00 X 04 10 X 07 01 3 09 10 1 13 01 0 16 10 1 01 03 1 06 °S X IZ °5 1 you for this SIR, Tour Mo ft Affectionate Correfpondent Bank of England, O&ob. q x S l 7 1 $' Fellow-labourer. Arenasius, ——- I I' -- r. X , J - •A* . f - ' ■■ •' .*} X\ A A • o c > c : - X "'X 4 •> ■ ; c ., . t c: : *:■ x o CP r ; . O ■■ Cl CX - - - -— '- i ..». j O-fi ■ Cl 1 3 l- 1 CX • — . — - ;> c X Cl ' l I C * • ■■--- £ : ■ f :i V -— - o -C r. 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