Page:Cyclopaedia, Chambers - Volume 1.djvu/867

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GAM

GAMELIA, Nuptial Fcafts, heLJ^among the Antient Greeks, in the Month Gamelion.

They were thus called from yaa©-, Marriage ; whence ra/*»Ai©-, an Epithet, or Surname, given to Jupiter aftd jfunO) confidcr'd as presiding over Marriages.

GAMELION, orGAMELiuM, is a Poem, or Compofi- tion in Verfe, on the Subject of a Marriage; more ufual y call'd Epithalamium. See Epithalamium.

GAMING, the Art, or Ad of performing, or pra&ifing a Game; particularly a Game of Hazard. See Game.

All publick Gaming is ieveriiy prohibited ; and what Mo- ney is thus loft, is recoverable again by Law. See Play.

In Cbina t Gaming is equally prohibited the common Peo- ple, and the Mandarins; and yet this does not hinder their Playing, and frequently lofing all they have ; their Lands, Houfes, Children, and even Wife, which are all fometimes, laid on a fingle Card. F. ie Comte.

The Bufinefs of Chance, or Hazard, is of Mathematical Confideration; inafmuch as it admits more, and left. It is, or is fuppos'd to be, an Equality of Chance; upon which the Gamefters fet out : This Equality is to be broke in upon in the Courfe of the Game by the greater good Fortune, or Adrefs of one of the Parties; upon which he comes to have a better Chance ; To that his Share in the De- posit, or Stakes, is now proportionably more, or better than at firft : This more and Ids is continually varying, and runs thro' all the Ratio's between Equality, and infinite Dif- ference; or from an infinitely little Difference till it arrives at an infinitely great one, upon which the Game is ended. The whole Game, therefore, with refpecT: to the Event or Iffue thereof, is only a Change of the Quantity of each Perfon's Share, or Chance ; or of the Proportion their two Shares bear to each other ; which Mathematicks alone can meafure. See Chance.

Hence ieveral Authors have computed the Variety of the Chance in feveral Cafes and Circumttances that occur in Gaming; particularly M. de Moivre, in a TreatiG',, De Menfnra Sortis : Which, as it may either be of fervice to the practical Gamejier, or the Setter, in caching them on what Side the Advantage lies; and whether they may lay on the fquare; or to the ipeculative one, in letting him in- to the way of thinking and determining in fuch cafes, we ihall here give the Reader an Abltract thereof.

Laws of Chance applied to Gaming.

Suppofe p the Number of Cafes wherein an Event may happen, and q the Number of Cafes wherein it may net happen; both Sides, the contingent and non-contingent, have their Degree of Probability ; And if all the Cafes wherein the Event may, and may not happen, be equally eafy ; the Probability of the happening, to that of the not- happening, will be as p to q.

If two Gamefters, A and B, engage on this footing, that if the Cafes p t happen, A mall win - but if q happen, B ihall win ; and the Stake be a ; The Chance, or Expectancy of A pa q a

will be ; and that of B ; confequently if A or B

. t-\~H t~Vq p a

fell their Expectancies, they .mould have for them ■

q a p\q

and refpeclively.

t-u

If there be two independent Events ; and p be the Num- ber of Cafes wherein the firft may happen ; and q the Kumber of thofe wherein it may not happen 5 and r the Number of Cafes wherein the fecond Event may happen, and s the Number of thofe wherein it may not happen; multiply p\-q by r-\-x ; the Product:, viz. p rA-q r-\-f sA-q s will be the Number of Cafes wherein the Contingency or non-Contingency of the Events maybe varied.

Hence, if A lay with B, that both Events mail happen, the Ratio of the Chances will be as p r to q r-\~p S-\-q s. Or if he lay that the firft ihall happen, but not the fecond ; the Ratio of the Chances will be as p s to f r-\~q r-\-q s. And if there were three, or more Events ; the Ratio of the Chances would be found by Multiplication alone.' —

If all the Events have a given Number of Cafes wherein they may happen, and alfo a given Number of Cafes where- in they may not; and a be the Number of Cafes wherein any one may happen, and b the Number of Cafes wherein it may not; and n be the Number of all the Events : Raife aA-b to the Power of n.

If now A and B agree, that if one or more of the E- vents happen, A Ihall win ; if none, B : The Ratio of the Chances will be as a-|-J|» — b" to b'> ; for the only Term where a is not found, is &"•

Trob. I- If J and 3 flay with a Jingle Die, on this condition, tbat if A thro-w two or more Aces at ei%ht throws, he flail win ; othcrwife % flail win : What is the Ratio of their Chances ?

Sol. Since there is but one Cafe wherein an Ace may turn up. and 5 wherein it may riot ; lettf— 1, and£— 5. And again*, fince there are eight Throws of the Die, let ?z— S ; and you

will havca-l-£|" — b' — n ab'— •' to b'.\-nab' 1. That

is, The Chance of J, will be to that of S, as o^pi, to 101 5625 ; or nearly as 2, to 3.

Trob. II. A and S are engaged at Jingle Qtioits, and. after flaying fame 'Time, A wants 4 of being up, aiii S, 6 ; but S is fo much the better Gamejlcr, that his Chance againjl A upon a fingle Throw, would be as 3 to 2 .- What is the Ratio of their Chances ?

Sol. Since A wants 4, and B, g, the Game will be ended in 9 Throws at the moft ; therefore raife a-\-b to the ninth Power, and it will be a ' a A- 9a' b-\-^6 a bb-\-%4-a 6 b s_|_ 126 ajb*-\-n6 a* b'Ar^a' b 6 -\-s6 a ab 7 -\-9ab s -\-b* : And take all the Terms wherein a has 4 or more Dirnen- fions, for A ; and all thofe wherein it has 6 or more, for B : And the Ratio of the Chances will be as a'-|-9 a'b-\-z6ai bb-\-$4a 6 b'A-iz6 a 5 £«_|-i2<> s 4 i 5, to 84 a % V-\-%6 aab'l. 9 a b*A-bi. Call a, 3 ; and b> 2 ; and you will have the Ra- tio of the Chances in Numbers 1759077 to 194048.

Trob. III. A and S are to flay with fingle Quoits ; and A is the heft Gamejler ■ fo that he can give S, z in % gj- Wbae is the Ratio of their Chances, then, in a fingle Throw ?

Sol. Suppofe the Chances as z to I ; and raife z-\- 1 to its Cube ; it will be z'A\-izzA-lz-\-l. Now fince A could give B 2 out of 3 ; A might undertake to win three Throws running ; and confequently the Chances in this Cafe will be as z 3 to 32i5~|-3s-l-i. Confequently, s J — 33E-I-3S-I-1. Or,

2-I-I; andcon-

-2z-\-i. And therefore z ,

The Chances therefore are ^ '-

zz'^zz'A-S'

_j

fequently z— J ' % - and 1 refpeftively.

Trob. IV. To find at how many Trials it is probable any Event will happen ; fo that A and S may lay a Wager iipon even Terms.

So'.. Let the Number of Cafes, wherein the thing may happen at the firft Trial, be a, thofe wherein it may not,£- and x the Number of Trials, wherein it is an even Chance, whether the Thing happen or not. By what is above fhcwa a4-j|* — Z>*r=£*.- Or, aA-b\* = i.b"- Therefore, x^=

V?* *' ■ ■ Again, refume the Equation aA-b'i * —

Log. aA'b — Log. b

2 b*, an d let a : b : : 1 .- q, and the Equation will change

1 IX 1

into this 1 -| =2. Raife I -|— to the Power of *,

q\ q x X X — I

by Sir I. Newton's Theorem, and let 1 -\ ~\- — x ■

1 J lit. x x- — I x — 2

-1 X X, &C. — t.

12 iq>

In this Equation, therefore, if q— 1, then x—s : Kq b& infinite, x will alfo be infinite. Suppofing x, to be infinite,

X XX X s

the Equation above will be 1 4- ' — ' -1- ' — ™ 4 ~ ' " " '1 & c - :=a •

X q iqq 6 q*

Again, let — = s, and we fliall have iA-z-\- ± zz -\- £ z 3

1 £=?c. =2. But i-l-z-l- |-SK>-|- ^s ! ,&c. iSc. is a Number whofe hyperbolical Logarith. is z ; confequently 3 =: Log. 2. But the hyperbolical Logarithm of 2 is 7 very nearly ; and therefore a— 7 nearly.

Hence where q is r, there #— 1 q ; and where q is infi- nite, a,"=7 q nearly. Thus are the Limits of the Ratio of x to q fix'd ; for that Ratio begins with Equality, and when rais'd to Infinity, ends at length in the Ratio of 7 to i» nearly.

Examp. 1. To find in how many Thro-i take to throw two Aces with two Tiice ?

A may

Sol. Since A has but one Cafe wherein he may throw two Aces with two Dice; and 35 wherein he may not, $=.35: Therefore multiply 35 by .7 ; the Produft 24. 5. fhews that the Number of Throws required is between 24 and 25.

Lem. To find the Number of Cafes, wherein any given Number of Toints may be thrown with a given Number of Dice?

Sol. Let fA-i be the given Number of Points ; n the Number of Dice ; and / the Number of Sides or Faces of each Die: Let f~f=q, q—f—r, r—f—s, s-^f-^t, & c - The Number of Cafes requir'd will be,

p f—i f

A,- — x x •

J 2

—, &.