to gain one Game before another gains two, may wager 3 to 1.

Prop. V. Suppose I want but one Game, and my Fellow Game/ter three, it is required to make a juft Diftribution of the Stake."

L

ET us here likewife confider in what ftate we fhould be, if I or he gain'd the firft Game ; if I gain, I have the Stake a, if he, then he wants yet 2 Games, and I but 1, and therefore we fhould be in the fame Condition which is fuppofed in the former Propofition; and fo there would fall to my Share, as was demonftrated there, a; therefore with equal facility there may happen to me a, or 3a, which, by the firft Propofition, is worth 3a, and to my Fellow-Gamefter there is left ja, and therefore my hazard to his is as 7 to 1.

As the Calculation of the former Propofition was requifite for this, fo this will ferve for the following. If I fhould fuppofe myself to want but one Game, and my Fellow four, (by the fame Method) you will find of the Stake belongs to me, and to him.

PROP. VI. Suppose I want two Games, and my Fel5 at low Gamefter three.

T

HEN by the next Game it will happen that I want but one, and he three, which (by the preceeding Propofition) is worth a; or that we fhould 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 fa ora, which by the first Propofition is worth it; and fo there are eleven parts of the Stakes due to me, and five to my Fellow.

PROP. VII. Let us fuppofe I want two Games, and my Fellow four.

F I gain the next Game, then I fhall want but one, and my Fellow four; but if I lofe it, then fhall want two, and he three: So I have an equal hazard for gaining a, or Ha, which, by the first is worth a: So it appears, that he who is to gain two Games for the other's four, is in,a better Condition than he who is to gain one for the other's two; for fhare in the first cafe is a or a, which is less than T, my fhare in the last.

my

PROP. VIII. Let us fupppose three Gamesters, whereof the firft an fecond want Game, but the third

2.

T

O find the fhare of the firft, we must confider what would happen if either he, or any of the other two gain'd the first Game; if he gains, then ne 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 a would be due to every one of them. Thus the firft Gamefter has one Expectation to gain a, one to gain nothing, and one for a, (fince all are in equal probability to gain the firft Game) which by the fecond Propofi tion is worth a: Now fince the fecond Gamefter's Condition is as good, his Share is likewife a, and fo there remains to the third a, whofe Share might have been as eafily found by itself.

PROP. IX. In any number of Gamefters you pleafe, amongst whom there are fome fewer Games: To find what is any one's share in the Stake, we must confider what wonld be due to him, whofe Share we inveftigate, if either he, or any of his Fellow-Gamefters fhould gain the next following Game; add all

their Shares together, and divide the Sum by the number of the Gameflers, the Quotient is his Share you were feeking.

SUPP

Uppose three Gamefters A, B, and C; A wants 1 Game, B 2, and C likewife 2, I would find what is the fhare of the Stake due to B, which I

fhall call

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 Bhimself gain, then he wants 1 Game, A1, and C2; therefore, by the former Propofition, there is due to him in that Cafe 49, then if C gains the next Play, then A and C would want but 1, and B 2; and therefore, by the eighth Propofition, his Share would be worth 4; add together what is due to B in all thefe three Cafes, viz. oq, the Sum is 39, which being divided by 3, the number of Gamefters, gives, which is the Share of B fought for: The Demonftration of this is clear from the fecond Propofition, because B has an equal hazard to gain og org, that is ++39, i. e. 9: Now it's e

3

vident the Divifor 3 is the number of the Gamefters.

To find what is due to one in any Cafe; viz. if either he, or any of his Fellow Gamefters win the following Game; we must confider firft the more fimple Cafes, and by their help the following; for as this Cafe could not be folved before the Cafe of the eighth Propofition was calculated, in which, the Games wanting were 1, 1, 2; fo the Cafe, where the Games wanting are 1, 2, 3, cannot be calcula ted, without the Calculation of the Cafe, where the Games wanting are 1, 2, 2, (which we have juft now perform'd) and likewife of the Cafe, where the

Games

Games wanting are 1, 2, 2, (which we have just now perform'd) and likewife of the Cafe, where the Games wanting are 1, 1, 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 want.

3313 12, 3, 4 22.35

Their Shares 133,55.55 451,195,83 433,635,119

243

729 1187

As

As for the Dice; thefe Questions may be proposed, at how many Throws one may wager to throw 6, any Numher below that, with one Die; How many Throws are required for 12 upon two Dice; or 18 on 3; and feveral other Queftions to this purpose.

For the refolving of which, it must 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 exact figure of a Cube: On two Dice there are 36 different throws; for in refpect 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 in 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 number of Dice, taking always, for the Addition of a new Die, 6 times the number of the preceding,

Befides, it must be obferved, that in two Dice there is only one way 2 or 12 can come up; two ways that 3or 11 can come up; for if I fhall call the Dice A and B, to make 3 there may be rin A, and 2 in B, or 2 in A, and in B; fo to make 11, there may be 5 in A, and 6 in B, or 6 in A, or 2 as well in A as B; for 10 there are likewife three Chances ; for 5 or 9 there are four Chances; for 6 or 8 five Chances; for 7 there are fix Chances,

In