Reafoning, I can eftimate both the Value of his Expectation and mine, and confequently (if we agree to leave the Game imperfect) determine how great a fhare of the Stakes belong to me, and how much to my Play-fellow; or if any were defirous to take my Place, at any rate I ought to fell it. Hence may arife innumerable Queries among two, three, or more Gamefters: And fince the Calculation 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 fhew how it is to be done, and afterwards explain those things which belong properly to the Dice.

In both Cafes I fhall make ufe of this Principle, One's Hazard or Expectation to gain any thing, is worth fo much, as, if he had it, he could purchase the like Hazard or Expectation again in a just and equal Game.

For Example, if one, without my Knowledge, fhould 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 should give me 5 Shillings; because, if I have 5 Shillings, I can purchase as good a Chance again, and that in a fair and juft Game.

Prop. I. If I expect a or b, either of which, with equal probability, may fall to me, then my Expectation is worth a+b, that is, the half Sum of a and b..

T

HAT I may not only demonftrate, but likewife investigate this Rule, fuppofe the Value of my Expectation be x; by the former Principle having, 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 ftakes

*, 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 za if I again; for in this cafe I get 2x, which are the Stakes, out of which I must pay the other a; but if 2x-a were worth b, then I have an equal hazard to get a or b; therefore making 2x-ab,

2

which is the Value of my Expectation. The Demonstration is easy, for having +, I can

a+b

play with another who will stake against it, on

$2

this Condition, that the Gainer should give to the Lofer a; by this means I have an equal Expectation to get a if I lofe, or b if I win; for in the last cafe I get a+b the Stakes, out of which I must pay a to my Play fellow.

In Numbers: if I had an equal hazard to get 3 or 7, then by this Propofition, my Expectation is worth 5, and it is certain, having 5, I may have the fame Chance; for if I play with another, fo that every one ftakes 5, and the Gainer pay to the Lofer 3, this is a fair way of gaming; and it is evident I have an equal hazard to get 3 if I lofe, or 7 if I gain.

Prop. II. If I expect a, b, or c, either of which, with equal facility, may happen, then the Value of a+b+c my Expectation is or the third part of the Sum of a, b, and c.

F

3

OR the Investigation of which, fuppofe x be the value of my Expectation; then x must be fuch, as I can purchase with it the fame Expectation in a just Game: Suppofe the Conditions of the Game

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be,

be, that playing with two others, each of us ftakes x, and I bargain with one of the Gamesters, if I win, to give him b, and he shall do the fame to me; but with the other, that if I gain, I shall give him c, and vice versâ; this is fair play : And here I have an equal hazard to get b, if the first win, c if the second, or 3xbc if I gain myself; for then I get 3x, viz. the Stakes, of which I give the one b and the other; but if 3x-bc be equal to a, I have an equal Expectation of a, b, or c; therefore making 3xb = c = a; x = a+b+, which is the Value of

-3.

my Expectation. After the fame Method you will find, if I had an equal hazard to get a, b, c, or d, the Value of my Expectation a+b+c+d +, that is

4

the fourth part of the Sum of a, b, c, and d, &c.

Prop. III. If 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 pa + bq; i. c. the Product of a multiplied in the

number of its Chances added to the Product of b, multiplied into the number of its Chances, and the Sum divided by the number of Chances both of a and b.

Uppofe, as before, be the Value of

pectation; then if I have x, I must be able to purchase with it that fame Expectation again in a fair Game: For this I fhall take as many Play-fellows as, with me, make up the number of p +9, of which let every one fake x, fo the whole Stake will be px+qx, and every one plays with equal hopes of winning; with as many of my Fellow: Gamefters

I my Ex

Gamefters as the Number q ftands for, I make the bargain one by one, that whoever of them gains fhall give me b, and if I win, I fhall do fo to them; with every one of the reft of the Gamefters, whofe Number is p-1, I make this bargain, that whoever of them gains, fhall give me a, and I fhall give every one of them as much, if I gain: It's evident this is fair play; for no Man here is injur'd; and in this cafe I have a Expectations to gain b, and p-1 Expectations to gain a, and 1 Expectation (viz. when I win myfelf) to get px + qx bgapa; for then I am to deliver b to every one of the q Players, and a to every one of the p I Gamefters, which makes gb+paa; if therefore qx + bx — bq — ap + a were equal to a, I would have p Expectations of a (fince juft now I had p- Expectations of it) and q Expectations of b, and fo would have juft come to my first Expectation; therefore putting px+qx- bq — apt a = a, and then is x= ap + bq

In Numbers: If I had 3 Chances to gain for 13, and 2 for 8, by this Rule, my hazard is worth 11; for 13 multiplied by 3 gives 39, and 8 by 2, 16, thefe two added, make 55, divided by 5 is II; and I can eafily fhew, if I have 11, I can come to the like Expectation again; for playing with four others, and every one of us ftaking 11, with two of them Í make this bargain, that whoever gains fhall give me 8, and I fhall too fo to them; with the other two I make this bargain, that whoever gains fhall give me 13, and I them as much if I gain, and 3 Expectations 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

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two

two 8, and the other two 13, and fo there remains 13 for myself.

Prop. IV. That I may come to the Question propos'd, viz. The making a juft Distribution amongst Gamefters, when their Hazards are unequal; we must begin with the most eafy Cafes.

then I play with another, on condition

that he who wins the three firft Games fhall 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 fhare.

The first thing we must confider in such Questions is the number of Games that are wanting to both: For Example, if it had been agreed betwixt us, that he thould have the Stakes who gain'd the first 20. Games, and if I had gain'd already 19, and my Fellow-Gamefter but 18, my hazard 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 must confider what would happen if the Game went on; it is certain, if I gain the first 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 as but fince I have an equal hazard to gain or lose the firft Game, I have an equal Expectation to gain as or a, which, by the first Propofition, is as much worth as the half Sum of both; i. . 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