bability of gaining or lofing 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 12 at least at two Throws of two Dice, because 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 abovemention'd, he ought just to give 14 for it. The equal wager in one Throw of two Dice, is to throw 7 at least one time, and 8 at least another time, and fo per vices: The Reafon is, be caufe in the firft Cafe I have 21 Chances againft 15. and in the second 15 Chances against 21. Of RAFFLING. 3 18 Throws. Chan. I 4.17 615 7 14 813 9 12 IO II 3649946 N Raffling, the different Throws and their Chances are thefe; Where ir is to be observed, that of the 216 different Throws of three Dice, there are only 96 that give Doublets, or two, at least, of a kind; fo it is 4 to 5 that with three Dice you fhall throw Doublets, and it is 1 to 35 that you throw a Raffle, or all three of a Kind. It is evident likewise, that it is an even Wager to throw II or above it, because there are as many Chances for 11, and the Throws above it, as for the Throws below it; but tho' it be an even Wager to throw II at one Throw, it is a Difadvantage 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 shall throw 21 at two Throws in Raffling, because it is as probable that you will, as that you will not throw 11, at least, the first time, and more than probable that you will throw io, at least the Second Time. For For an inftance of the plainnefs of the preceding Method, I will fhew, how by fimple Subtraction, the moft part of the former Problems my be folv'd. Suppofe A and B, playing together, each, of them takes 32 Shillings, and that A wants one Game of the number agreed on, and B wants two; to find the fhare of the Stakes due to each of them. It's plain, if A wins the next Game, he has the whole 64 Shillings; 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 fure 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; therefore fays. A to B, give me the 48 which I am sure 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 8, which, with the former 48, 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 against it, one Throw for 6 is worth of the Stake; therefore fays B to A, of the 216 Guineas take a fixth part for your firft Throw, that is, 36; for your next Throw take a fixth part of the remaining 180, that is, 30 ; and for your third Throw, take a fixth part of the remaining 150, that is, 25, D 4 which which in all make 9 1, leaving to me 125; fo his hazard who undertakes to throw 6 at one of the three Trows, 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, every Throw for 6 of one Die, is worth the fixth part of what I throw for; therefore for my firft Throw give me 216, which is the fixth part of 1296, and there remains 1080, I must have the fixth part of that, viz. 180, for my fecond Throw; and the fixth part of the remaining 900, which is 150, for my third Throw; and the fixth part of the laft remainder 750, which is 125 for my fourth; all this added together makes 671, and there remains to you 625; fo it is evident, that d's Hazard, in this Cafe, is to B's 671 to 525. Suppose A is to win the Stakes (which we shall 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; fays B to A, the Chances which give 7 are 6 of the 36, which is as much as 1 of 6; therefore for your first Throw you fhall 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; this in all makes 11; fo you leave 25 to me; fo A's Hazard is to B's as 11 to 25. It were eafy, at this rate, to calculate the moft. intricate Hazards, were it not that Fractions will occur; which, if they be more than 1, may be fuppos'd equal to an Unit, without caufing any remarkable Error in great Numbers. It will not be amifs, before I conclude, to give you a Rule for finding in any number of Games the Value of the firft, becaufe Huygens's Method, in that Cafe, is fomething tedious. Suppofe Suppofe A and B had agreed, that he should have the Stakes who did win the firft 9 Games, and A had already won one of the 9; I would know what fhare of B's Money is due to A for the Advantage of this Game. To find this, take the first eight even Numbers 2, 4. 6, 8, 10, 12 14, 16, and mul tiply them continually, that is, the firft by the fecond, the product by the third, &c. take the first eight odd Numbers 1, 3, 5, 7, 9, 11, 13, 15, and do just fo by them, the product of the even Number is the Denominator, and the product of the odd Number the Numerator, of a Fraction, which expreffeth the quantity of B's Money due to 4 upon the winning of the firft Game of 9; that is, if each ftak'd a number of Guineas, or Shillings, &c, exprefs'd by the Product of the even Numbers, there would belong to A, of B's Money, the Number exprefs'd by the product of the odd Numbers. For Example, Suppofe A had gain'd one Game of 4, then by this Rule, I take the three firft even Numbers, 2, 4, 6, and multiply them continually, which make 48, and the first three odd Numbers, 1, 3, 5, and multiply them continually, which make 15; fo there belongs to A of B's Money, that is, if each stak'd 48, there would belong to A, befides his own, 15 of B's. Now by Huygens's Method, if A wants but three Games while Bwants four, there is due to A of the Stake; by this Rule there is due to 4 of B's Money, which is of the Stake, which, with his own of the Stake, makes or of the Stake; and fo in every Cafe you will find Huygens's Method, and this will give you the fame Number: A Demonftration of it you may fee in a Letter of Monfieur Pascal's to Monfieur Fermat; though it be otherwife exprefs'd there than here, yet the confequence is eafily fupply'd. Te 4 prevent prevent the labour, of Calculation, I have fubjoin'd the following bale, which is calculated for two Gamsters, as Monfieur Huygens's is for three. If each of us stake 256 Guineas in There belongs to me 256 of my The Use 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,obf the first two. Games, ¿ 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' r, I fay, As 256vis tol96, fous pto al fourth.dm Vismat edi |