GAM
( J 22 )
GAM
Laftly, making fl=4, £=r8, M— n ; and the General Series will change into the following one T % -j- T | 'p -\- -^ JP
Or into this other ( by multiplying all the Terms by fome common Number judg'd moft expedient for the throwing out of Fractions, viz. in the prefent Cafe, by 495)
16*5 -|- 120-]- 84 -1- 5*4" 3 5-1- 20 -I- 10 -1-4- 4- 1
And therefore A will have 165 -|- 56 ~\- 10—231 5 B will liave 120-1-35-1-4—159$ and G will have 84-|-20-\-i=ie5. So that their fever - al Expectancies will he as 231, 159, 10*3 or as 77, 53, 35.
- prob. IX. A and S having 12 Counter s, four of them
white, and eight black 5 A wagers with S, that tak- ing out fevcn Counters, blindfold, three of them fh all' lie -white; What is the Ratio of their Expectancies ?
Sol. i° Seek how many Cafes there is forfeven Counters, to be taken out of 1 2 5 they will be found from the Do&rine o£ Combinations, to be 792.
J | x '4 x *f x I x I x £ x 4= 792-
2 Set afide three white ones, and find all the Cafes wherein 4 of the eight black ones may be combined there- with ; they will be found to be 70.
And fince there arc four Cafes, in which three white may be taken out of four j multiply 70 by 4 : Thus, the Cafes, wherein 3 Whites may come out with 4 Blacks, are found to be 280.
3 Ey the common Laws of Gaming, he is reputed Con- queror, who produces an EffccT: oftener than he undertook to do, unlefs the Contrary be exprefly agreed on ; and therefore, if A take out 4 Whites wirh 3 Blacks, he wins. Set afide 4 Whites, then, and find all the Cafes wherein 3 of the 8 Bl;-.cks may be combined with four Whites: Thefe Cafes will appear to be 56".
|- x I x f — 5c?
4 A, therefore, has 280-I ^—336 Cafes, wherein he may win 5 which fubtracted from the whole Number of Cafes 792, leaves 456 the Number of Ca'es wherein he may lofe. The Ratio of the Chance of A, therefore, to that of S, is as 33<? to 45^5 or as 14 to 19.
To avoid too much Prolixity in this Article, we mult de- fift from further Investigations, which in the following Pro- blems grow very long, and more perpIexM. In the reft, therefore, wc fhall content ourfelves to give the Anfwer, or Refult, without the Procefs of arriving at it; which may be of ufe, as it furnifties fo many Data, from whence, as Standards, we may be enabled occasionally to judge of the Probability of Events of the like Kinds : Tho', without let- ting the Mind into the precife Manner, and Reafon thereof.
^Prob. X. A and & play with two Z)ice on this Condi- tion, that A fhall win, if he throw fix ■ and 2?, if he throw feven : A to have the firft 'Throw, in Hen of which 2? to have two Throws 5 and hcth to continue with t-wo Throws each turn, till one of them wins ; What is the Ratio of the Chance of A to that of .2? ?
Anf. As 10355 to 12276".
Prob. XI. If any Number of Gameflers, A, 5?, C, 7), E, &c. equal in point of ^Dexterity, depojit each one piece of Money, and engage on thefe Conditions, that two of them, A and S, beginning the Game ; which ever of them JJyall be overcome, fhall give place to the third, C, who is to play with the Conqueror 5 and the Conqueror here, to be taken up by the fourth Man, tD, and thus on; till fome one, having conquer 'd them all round, draws the Stake : What is the Ratio of their Expectancies ?
Sol. This Problem, M. Bernoulli folves analytically. Here, calling the Number of Gameflers K-j-r, he finds that the Probabilities cf any two immediately following each other in the Courfe of playing, are in the Ratio 1-I-2" to 2 s 5 and therefore the Expectancies of the feveral Gameftcrs A B C D E &c. are in a Geometrical Progrefficn i-|-2" : 2*
- : a : c : : c : d : : d : e Sic.
Hence it is cafy to determine the State of the Probabili- ties of any two Gameflers, either before the Game, or in the Co^rfc thereof. If e. gr. there be three Gamefters, A, B, C, then n — 2 and i-|-2" : i n -. -. 5 : 4 : : a : c : That is, Their feveral 'Probabilities of winning, before A have overcome 2?, or '£, C ; are as the Numbers 5, 5, 4 j and therefore tl'eir Expectancies are T t, T £', _*. ; For all of them, taken together, muft make i> or t'.bfolutc Certainty. After
A has overcome $, the Probabilities of A, % and C --mil be ■j, ~, f, as in the Anfwer above. If there be four Game- iters, A B C D, their Probabilities from the Beginning will be as 81, 81, 72, 64. After A has beat B, the feveral Probabilities of B D C A, will be as 25, 32, 315, 56 re _ fpe&ively. After A has beat B and C, the Probabilities of C B D A will be as 16, 18, 28, 87.—
Prob. XII. Three Gamcjlers, A, S and C, whofc Z)exte. ■fities are equal, depofip each one Piece, and engage npon thefe Terms, That two of them fhall begin to play, and that the vanquiffjd Tarty fhall give place to the third, who is to take tip the Conqueror; and the fame Condition to go round j each Perfon when va?iqmfi'd, forfeiting a certain Sum to the main Stake ; which fhall be all fwept by the Perfon who firfl beats the other two fuccejfively. Hew much, now, is the Chance of A and S better or worfe, than that ofC\
Anf. i° If the Forfeiture be to the Sum each Perfon firft depofited, as 7 to 6, the Gamefters are upon an equal- footing. 2 If the Forfeiture be in a lefs Ratio to the De- poilt, A and B are on a better footing than C : If in a greater Ratio, the Advantage is on the Side of C. 3 Af- ter A has overcome B once, the Probabilities are as *£, *, 4 j or as 4, 2, 1 j viz. that of A the greateft, and of B the leaft.
M. 'Bernoulli gives an analytical Solution of the fame Problem, only made more general ; as not being confined to three Gamefters, but extending to any Number at plea- fure.
Prob. XIII. A and 23, two Gameflers cf equal dexte- rity, play with a given Number of Satis - 7 and after ■ fome time A wants 1 of being tip, and fi, 3 : What is the Ratio of their Chances ?
Anf. AV Expectancy is worth £ of the Stake, and Ws only £5 fo that their Chances axe as 7 to 1.
Trob. XIV. T:vo Gameflers, A and 2?, cf eqval tDexte- rity, arc engaged in play, on this Conditun, then as often as A exceeds S, he (hall give him one '-Piece of M^Vty ; and that "B '(J->a!l do the like, as oft as A ex- ceeds him; and that they Jhall not leave off, till t..e has won all the other's Money : Each now having four ^Pieces ; two Syfianders, R and S, lay a wa^ir en the Number of Turns, in which the Game Jhall be finijh'd ; viz. R, that it Jhall be over in 10 Turns ; What is the Value of the Expectancy of S.
Anf. r l4l- or ||- of the Wager j or it is to that of R as 560 to 464.' — ■
If each Player had 5 Pieces, and the WVger were, that the Game fhould end in ten Turns, and the Dexterity of A were double that of B; the Expectancy of S wou.d be
if each Gamefter have 4 Pieces ; and the Ratio of the Dexterities be required to make it an even Wager that the Game mall end in 4 Turns: It will be found that the one mult be to the other as 5.274 to 1.
If each Gamefter have 4 Pieces, and the Ratio of their Dexterities be requir'd to make it an even Lay that the Game mall be ended in 6 Turns 5 the Anfwer will be found to be, as 2.57(5' to 1.
%>rob. XV. Two Gameflers, A and 3, of equal dexte- rity, being agreed not to leave off playing till ten Games are over ; a Spectator, R, lays a Wager with another, S, that by that time, or before, A jhall have beat % by three Games : What is the Value of the Expectancy cf R ?
Anf-
572.
GAMM, Gammut, Gamut, or Gam-?//, in Mufic, a Scale, whereon we learn to found the Mufical Notes, lit, re, mi, fa, fol, la, in their feveral Orders, andDilpofitions. See Note, and Scale.
The Invention of this Scale is owing to Guido Aretin, a Monk of Arctium, in Tufcany 5 tho' it is not properly an Invention, as an Improvement on the Diagramma or Scale of the Antients. See Diagram.
The Gammut is alfo call'd the Harmon ical Hand; by rea- fon Guido firft made ufe of the Figure of the Hand, to ar- range his Notes on.
Finding the Greek Diagramma of too fmall Extent, Guido added five more Chords, or Notes to it : One, below the Proflambanomenos, or graveft Note of the Antients ; and four, above the Nete, or Acuteft. The firft, he call'd Hy- po-proflambanomenos j and denoted it by the Letter G, or rather the Greek V->Gammai Which Note being at the head^
or
- of the Wager 5 or it is to that of S as 3 52 to
