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A25748 Of the laws of chance, or, A method of calculation of the hazards of game plainly demonstrated and applied to games at present most in use : which may be easily extended to the most intricate cases of chance imaginable. Arbuthnot, John, 1667-1735. 1692 (1692) Wing A3602; ESTC R31565 22,800 122

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they have an equal Hazard to get the first of the four or miss it if they get it then they want two of the three and consequently there is due to them ½ of the Stake if they miss it then they want three of the three and consequently there is due to them 1 8 of the Stake therefore by Prop. 1. their Hazard is worth 5 16 leaving to C and D 11 16. A and B playing at Whist against C and D A and B have eight of ten and C and D nine and therefore can't reckon Honors to find the proportion of their Hazards There is 5 16 due to C and D upon their hazard of having three of four Honours but since A and B want but one Game and C and D two there is due to C and D but ¼ or 4 16 more upon that account by Prop. 4. this in all makes 9 16 leaving to A and B 7 16 so the hazard of A and B to that of C and D is as 9 to 7. In the former Calculations I have abstracted from the small difference of having the Deal and being Seniors All the former Cases can be calculated by the Theorems laid down by Monsieur Hugens but Cases more compos'd require other Principles for the easie and ready Computation of which I shall add one Theorem more demonstrated after Mons. Hugens's Method Theor. If I have p Chances for a q Chances for b and r Chances for c then my hazard is worth that is a multiplied into the number of its Chances added to b multiplied into the number of its Chances added to c multiplied into the number of its Chances and the Sum divided by the Sum of Chances of a b c. To investigate as well as demonstrate this Theorem suppose the value of my hazard be x then x must be such as having it I am able to purchase as good a hazard again in a just and equal Game Suppose the Law of it be this That playing with so many Gamesters as with my self make up the number p+q+r with as many of them as the nnmber p represents I make this bargain that whoever of them wins shall give me a and that I shall do so to each of them if I win with the Gamesters represented by the number of q I bargain to get b if any of them win ann to give b to each of them if I win my self and with the rest of the Gamesters whose number is r − 1 I bargain to give or to get c after the same manner Now all being in an equal probability to gain I have p Chances to get a q Chances to get b and r − 1 Chances to get c and one Chance viz. when I win my self to get px+qx+rx − ap − bq − rc+c which if it be suppos'd equal to c then I have p Chances for a q Chances for b and r Chances for c for I had just now r − 1 Chances for it therefore if px+qx+rx − ap − bq − rc+c = c then is By the same way of reasoning you will find if I have p Chances for a q Chances for b r Chances for c and s Chances for d that my hazard is c. 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 for and it is easie to prove that with 5 Shillings I can purchase the like hazard again for suppose I play with six others each of us staking 5 Shillings with two of them I bargain that if either of them win he must give me 3 Shillings and that I shall do so to them and with the other four I bargain just so to give or to get 5 Shillings This is a just Game and all being in an equal probability to win 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 self for then out of the Stake which makes 35 Shillings I must give the first two 6 Shillings and the other four 20 Shillings so there remains just 9 to my self It it easie by the help of this Theorem to calculate in the Game of Dice commonly call'd Hazard what Mains are best to sett on and who has the Advantage the Caster or Setter The Scheme of the Game as I take it is thus   Throws next following for Mains The Caster The Setter V. V. II. III. XI XII VI. VI. XII XI II. III. VII VII XI XII II. III. VIII VIII XII XI II. III. IX IX II. III. XI XII By an easie Calculation you will find if the Caster has VI. and the Setter VII there is due to the Caster ⅓ of the Stake if he has V. against VII 2 5 of the Stake VI. against VII 5 11 of the Stake IV. against VI. 3 8 of the Stake V. against VI. 4 9 of the Stake VI. against V. 3 7 of the Stake I need not tell the Reader that IV. is the same with X V. with IX and VI. with VIII Suppose then VII be the Main To find the proportion of the hazard of the Caster to that of the Setter By the Law of the Game the Caster before he throws next has four Chances for nothing viz. these II III XII eight Chances for the whole Stake viz. those of VII XI six Chances for ⅓ viz. those IV X eight Chances for 2 5 viz. those of V IX and ten Chances for 5 11 viz. these of VI X so his hazard by the preceding Theorem is Now to save the trouble of a tedious reduction Suppose the Stake which they play for be 36 that is the Setter had laid down 18 in that case every one of these Fractions are so many parts of an Unite which being gather'd into one Sum give 1741 59 to the Caster leaving 1814 55 to the Setter so the hazard of the Caster is to that of the Setter 244 251. Suppose VI. or VIII be the Main then the Share of the Caster is II. III. VI. IV. V. XI XII X. IX VIII VII 5×0+6×1+6×3 8+8×4 9+5×½+6×6 11 = = 17229 396 leaving to the Setter 18167 396 so the hazard of the Caster is to that of the Setter as 6961 to 7295. Suppose V. or IX be the Main then the Share of the Caster is II. III. XI IV. VI. XII V. X. IX VIII VII 6×0+4×1+6×●+4×½+10×●+6×● = = 17 229 315 leaving to the Setter is 1886 315 so the hazard of the Caster is to that of the Setter as 1396 to 1493. It is plain that in every Case the Caster has the Disadvantage and that V. or IX are better Mains to set on than VII because in this last Cast the Setter has but 18 and 14 55 or 84 330 whereas when V. or IX is the Main he has 1886 315 likewise VI. or VIII are better Mains than V. or IX because 167 396 is a greater Froction than 86 315. All those Problems suppose Chances which are in an equal probability to happen if it should be suppos'd otherwise there will arise variety of Cases of a quite different nature which perhaps 't were not unpleasant to consider I shall add one Problem of that kind leaving the Solution to those who think it merits their pains In Parallelipipedo cujus latera sunt ad invicem in ratione a b c Invenire quotâ vice quivis suscipere potest ut datum quodvis planum v. g. ab jaciat FINIS ERRATA PReface page 3. line 1. read in p. 6. l. 5. r. incur p. 10. l. 8. for is left to me r. properly deserves the name of Conduct Book p. 2. l. 7. for 9 r. q. p. 16. l. 5. add and he one p. 71. l. 5. r. wins Advertisement THe whole Duty of Man according to the Law of Nature By that famous Civilian SAMUEL PUFFENDORF Professor of The Law of Nature and Nations in the University of Heidelberg and in the Caroline University afterwards Counsellour and Historiographer to the K. of Sweden and to his Electoral Highness of Brandenburg Now made English Printed for C. Harper at the Flower-de-Luce over-against St. Dunstan's Church in Fleetstreet