Prop. X. To find at how many times one may undertake to throw 6 with one Die.

I'

F any fhould undertake to throw 6 the first time, it's evident there's one Chance gives him the Stake, and five which give him nothing; for there are 5 throws against him, and only one for him. Let the Stake be called a, then he has one Expectation to gain a, and five to gain nothing, which, by the third Propofition, is worth a, and there remains for the other sa; fo he who undertakes, with one Die, to throw 6 the first time, ought to wager only I to 5.

2. Suppofe one undertake, at two Throws of I Die, to throw 6, his Hazard is calculated thus; if he throw 6 at the firft, he has a the Stake; if he do not, there remains to him one throw, which, by the former Cafe, is worth a; but there is but one Chance which gives him 6 at the fift throw, and five Chances against him; fo there is one Chance which gives him a, and five which give a, which by the fecond Propofition, is worth a, fo there remains to his Fellow-Gamefter a; fo the Value of my Expectation to his, is as 11 to 25, i. e. less than i

to 2.

By the fame Method of Calculation, you will find, that his hazard who undertakes to throw 6 at

three

three times with one Die, isa; fo that he can only lay 91 against 125, which is fomething lefs than 3 to 4.

He who undertakes to do it at four times, his haz. ard is, fo he may wager 671 against 625, that is, fomething more than I to 1.

He who undertakes to do it at five times, his hazard is a, fo he can wager 671 against 625, that is, fomething less than 3 to 2.

His hazard who undertakes to do it at 6 times, is 110314, and he can wager 4651 against 3125, that is, fomething less than 2 to 1.

Thus any number of throws may be eafily found; but the following Propofition will fhew you a more compendious way of Calculation.

Prop. XI. To find at how many times one may undertake to throw 12 with two Dice.

Fone fhould undertake it at one throw, it's clear he has but one Chance to get the Stake a, and 35 to get nothing; which, by the third Pro pofition, is worth.

He who undertakes to do it at twice, if he throw 12 the first time, gains a; if otherwise, then there remains to him one throw, which, by the former Cafe, is worth a; but there is but one Chance which gives 12 at the first throw, and 35 Chances against him; so he has 1 Chance for a, and 35 for a, which by the third Propofition is worth yoga, and there remains to his Fellow-Gamefter 1138a. From these it's eafy to find the Value of his hazard, who undertakes it at four times, paffing by his cafe who undertakes it at three times.

1225

If he who undertakes to do it at four times throws 12 the first or fecond Caft, then he has a; if not, there remains two other throws, which, by the

former

former Cafe, are worth a; but for the fame reafon, in his two first throws, he has 71 Chances which give him a, against 1225 Chances, in which it may happen otherwife; therefore at firft he has 71 Chances which give hima, and 1225 which give him 1234, which by the third Propofition is worth 438a, which fhews that their hazards to one another are as 178991 to 1500625.

178991

From which Cafes, it is eafy to find the Value of his Expectation, who undertakes to do it at 8 times, and from that, his Cafe who undertakes to do it at 16 times; and from his Cafe who undertakes to do it at 8 times, and his likewife who undertakes to do it at 16 times; it is easy to determine his Expec tation who undertakes to do it at 24 times: In which Operation, because that which is principally fought, is the number of throws, which makes the hazard equal on both fides, viz. to him who undertakes, and he who offers, you may, without any fenfible Error, from the Numbers (which elfe would grow very great) cut off fome of the laft Figures. And fo I find, that he who undertakes to throw 12 with two Dice, at 24 times, has fome lofs; and he who undertakes it at 25 times, has fome advantage.

Prop. XII. To find how many Dice one can under take to throw two Sixes at the firft Caft.

TH

HIS is as much, as if one would know, af how many throws of one Die, he may undertake to throw twice fix: now if any should undertake it at two throws, by what we have fhewn before, his hazard would be a; he who would undertake to do it at 3 times, it his first throw were not 6, then there would remain two throws, each of which must be 6, which (as we have faid) is worth ; but if the firft throw be 6, he wants

only

only one 6 in the two following throws, which by the tenth Propofition, is worth a but fince he has but one Chance to get 6 the first throw, and five to miss it; he has therefore, at first, one Chance fora, and five Chances fora, which, by the third Propofition, ie worth a, ora; after this manner still assuming I Chance more, you will find that you may undertake to throw two Sixes at 1ð throws of one Die, or throw of ten Dice, and that with fome advantage.

16

Prop. XIII. If I am to play with another one Throw, on this condition, that if 7 comes up. I gain, if 10 be gains; if it happens that we must divide the Stake, and not play, to find how much belongs to me, and how much to him.

B

Ecaufe of the 36 different Throws of the two Dice, there are fix which give 7 and 3 which give 10, and 27 which equals the Game, in which cafe there is due to each of us a: But if none of the 27 fhould happen, I have 6, by which I may gain a, and 3, by which I may get nothing, which by the third Propofition, is worth a; fo have 27 Chances fora, and 9 for 3a, which by the third Propofition, is worth 34, and there remains to my Fellow-Gamefter 1a.

Prop. XIV. If I were playing with another by turns, with two Dice, on this Condition, that if I throw I gain, and if he throw 6 he gains allowing him the first Throw: To find the proportion of my Ha zard to his.

SUPE

Uppofe I call the Value of my Hazard x, and the Stakes a, then his Hazard will be a---*; then wherever it's his turn to throw,myHazard is,

but

but when it's mine, the Value of my Hazard is greater. Suppofe I then call it y; now because of the 36 throws of two Dice, there are five which give my Fellow-Gamefter 6, thirty-one which bring it again to my turn to throw, I have five Chances for nothing, and thirty-one for y, which, by the third Propofition, is worth y; but I fuppos'd at firft my Hazard to be x; therefore, and confequently y=x. I fuppos'd likewife, when it was my turn to throw, the Value of my Hazard was y; but then I have fix Chances, which give me 7, and confequently the Stake, and thirty which give my Fellow the Dice, that is make my Hazard worth fo I have fix Chances for a, and thirty

for x, which, by Prop. 3. is worth 64+30%, but

36

this by fuppofition is equal to y, which is equal (by what has been prov'd already) to 2x; 36 therefore

30x+6α

36

36

31

and confequently xa, the Va

lue of my Hazard, and that of my Fellow-Gamefter is a, fo that mine is to his as 31 to 30.

Here follow fome Questions which ferve to exercife the former Rules.

I 1. A and B play together with two Dice, A wins if he throws 6, and B if he throws; A at first gets one throw, then B two, then A two, and fo on by turns, till one of them wins. I require the proportion of A's Hazard to B's? Anfwer, It is as. 10355 to 12276...

2. Three Gamefters, A, B, and C,, take 12 Counters, of which there are four white and eight

black;