GAM
( 121 )
GAM
afr. x — .
-t$C. X — X -
X x — — g?c. X — X — -
I 2 3
Which Series is to be continued, either become equal to nothing,
t -%
■ x — — 3
till fome of the Factors negative. And note, 1 fo many Factors of the fevcral Products — x ■
r r — l r — i s s — i I z 2
tSc. — x — — x —-£?<;.— x — f£c. are to be taken as
12 2 12
there are Units in » — i.
"Prax. Suppofe the Number of Cafes required wherein 16 "Points may be thrown with four 2) ice ?
•l-Vx'^Y = 455
— f x i x I x f — — 33<5
4- f x f x f x ? x i = -|- 6
Now, 455 — 3 3 <S —I— <S = 125 ; fo that 125 is the Number required.
Exatnp. 2. Jo jW at how many Cafts A may undertake to throw J 5 'Points with fix 'Dice ?
Sol. Since A has 1666 Cafes, wherein he may turn up 15 Points, and 44.990 againrt him; divide 44990 by 1666; and the Quotient 27 will be=<y. Therefore, multiply 27 by 7; the Product 18, 9 jhews the Number cf Thrones required to be nearly 19.
^Prob. V. To find the Number of Trials wherein it is probable any Event may happen twice ; fo that A and % may lay a Wager thereon with an equal Chance ?
Sol. Suppofe the Number of Cafes wherein the Event may happen the firft Trial, to be a ; and thofe wherein it may not, b ; and call the Number of Trials required, x : It appears from what is above ilicwn, that a-\- b\ = 2 "*
~\-zaxbx-—i. Or, making a : b :
IX
■4-' ■ 1° Lcty = i, and then x-
1 I 2 Let q be infi-
nite, and x will alfo be infinite : Suppofe x infinite, and x — z= S, and then 1 4. z + 1 » a 4- r * ! > S?c. = »4- * * 5
and therefore »=Zog. 2 4- -E°g. i+B : If then Zog. 2 be called jf; the Equation will be transform 'd into the follow-
& * ing fluftional one — = y. And inveffigating the Va-
lue of 3 by the Powers of y, we mail find 3— 1.578, near- ly - and therefore x will always be between the Limits 3 q and 1.678 q ; but a? will foon converge to 1.678 q ; and there- fore if q have nor a very fmall Ratio to 1, we may take xzrz 1.578 q. Or if there be any Sufpicion of * being too fmall,
[.578?
fubftitute its Value in the Equation 1 -|
= 2-1 , and
note the Error, if it be worth regarding ; Thus will x be a little increafed ; fubftitute the thus increas'd Value for x in the forcfaid Equation, and note the new Error : Thus, from the two Errors, may the Value of x be corrected with fufficient Accuracy.
Here we (hall add a Table of Limits, that will carry the
Intent of this Problem much further.' — ■
If the Wager be upon happening once, the Number of
Trials will be between
1 q and 0.693 ? If upon twice, between 3 q and 1.678 q
If upon trice, between 5 q and 2.675 ?
If upon four times, between 7 q and 3-67r q If upon five times, between 9 q and 4.673 q If upon fix times, between 1 1 q and 5.668 q
1 1 Points are thrown, A fiall give 33 one Piece ; dad every time 14 Points are thrown, S fhalL give A a •Piece ; and that hefhall win the whole, that firfl gets all the Money in his Hands : We demand the Ratio of the Chances of A to that of S.
Sol. Let p be the Number of Pieces each feverally takes *" and a and h the Number of Cafes wherein A and B may reflectively gain, each a piece; the Ratio of their Chances will be as a to h? : In this cafe, p~n, 0=27, £=15- or if when 27 : 15 : : 9 : ;. you make 0=9, £=5; and therefore the Ratio of the ExpcBancies will be as 9'* to 5'% or as 244140625 to 282429536481.
NS. Great Care muft be taken to avoid the confound- ing of different Problems together, from fome Appearance of Affinity between them. The following one feems very like the former.
Prob. VII. C having 24 Pieces, or Counters, throws three Dice; and every time 27 Points turn up, gives one Counter to A; and every time 14 turn up, gives one to S ; and A and S engage on this footing, that he who firfi gets 1 2 Counters, pall win the Stake : We require the Ratio of their Expectancies ?
This Problem differs from the preceding one, in that the Game muft neceffarily end in 23 Throws; whereas, in the former, it might hold out to Eternity, by reafon of the Re- ciprocations of Lofs and Gain, which deftroy one another.
Sol. Raife a -\- b to the 23' Power, and the 1 2 former Terms will be to the 12 latter as the Expectancy of A to that of B.
Prob. VIII. Three Gamefters, A S and C, have each twelve Sails, 4 of them white, and 8 black ; and be- ing hoodwinked, play on this condition, that the firfi who chufes a white Sail, fliall win the Stake ; and that A /hall have the firfi Choice, then S, then C ; and Co round again : What, then, is the Ratio of the Chances of A, S, C ?
Sol. Let n be the Number of Balls, a the Number of white ones, b of black ones, and n the Stake. Here
l° A, has the Cafes a, wherein he may chufe a white Ball ; and the Cafes b for a black one : Confcquently, his
a a
Expectancy, from the firft Choice, is — — , or ■ — . Where-*
a a\b 11
fore, fubtracting — from 1 5 the Value of the remaining n a n- — a b
Expectancies will be 1 '— — — * = — .
n n n
1° B, has the Cafes a for a white, and the Cafes b — 'i for a black one 5 but the firft Election is in A ; and 'tis un- certain, whether or no he may have won the Stake; and therefore the Stake, in refpect of B, is not i 3 but only b a
■ — ■ ; fo that his Expectancy from the firft Choice is ■
« b ab ab b a-\-b — t x — ss — — . Subtract — = from — and the Va- ra nxn — i nxn—i n v
nb — b — ab bx — 1
lue of remaining Expectancies will be = - __ ^ ,
n X n — 1 nxii — 1
3 C, has the Cafes a for a white ; / the Cafes b — 2 for a
black one ; and therefore his. Expectancy from the third
axbxb — 1
Choice is — . — =?
nxn — ixn — 2. 4 P After the like manner, A has the Cafes a for a white, and b — 3 for a black ; fo that at the fourth Choice, the Ex-
_ .,, , axbxb — ixb — 2 . „„ '
pettancy will be = . And lo of the reft.
nxn — 1x3 — -zxn — 3
a b b—i Write down, therefore, the Series — -|- ■ — . — P ~\ .
^.-1 A +1 J, Sec. Where P, Q_, R, Sy&ir. denofe the
» — 3 n — 4
preceding Terms, with their Characters ; and take as many Terms of this Series, as there are Units in b-\-x (for there cannot be more Choices than there are Units in b-|-l) and the Sum of all the third Terms, flopping the two Interme-
Examp. To find at how many Throws A may undertake diates, beginning from — , will be the whole Expectancy
to throw three Aces, twice, with three Dicel
Sol. Since A has but one Cafe, wherein he may throw three Aces ; and 21 5, wherein he may not ; ff=r2i5. There- fore multiply 215 by r.678. The Product 360, 7 will iliew the Number of Throws required to be between 36c and 361.
of A ; the Sum likewife of all the third Terms, commenc- 6
ing from p, will be the whole Expectancy of B ; and
n I h-*—\
the Sum of the Thirds, commencing from Q. the!
Prob. VI. A, and S dcpoflt each 12 pieces of Money, and whole Expectancy of C. play 'with three Dice, on this footing, that every time
- Hh
Laitly,
