1000000030

G A M

acratinn. Thus if I would know wh11Ihe probabililY

i! of milling anaee four lim ! IOgelher ..ith a die, thi!

eonfider

the failinK of fou r diffmnl erents. NolV

rhe probabiliry of milling rhe r,rH i!

rhe feeonu is

alfo

rhe rhiro

and rhe fou" h

rherefore rhe

prob"biliryof mi/ling it four times IOgether is

~X~Xi

X~=T~*;

which belng (ubtr.{t,d fr"m r, rhere \ViII

remain

TI-

for Ihe probabilil y of rhro\Ving il once or

oftener in four rimes: rherefore rhe odJs of Ihro"ing

ao aee in four rimes, is

625 .

Bur if rhe

flingin~

of "" ace \Vas u"dmaken in three

times, the

proh~bilily

oi milling il time limes would

bef'xix~={ ~ ~;

whieh being(ubttaéled from t, Ihm

IVil remlin,

for the

proba~ilitY

of rhrolVing Ir once

or oftener in time limes: thercfore the udds

a~ainfl

throwing it in time times are t

91.

Again , fu p.

llore IVe would knolV rhe probabililY of throwing

ace

once in four times,and no more: fince the prob.bility of

throwing it rhe firll time is

and of mllling it the o·

Iher rhree times is

~X¡Xi,

it follows thar the proba·

biliry of IhrolVing it Ihe firll rime . and mllling ir the

other time fucceflive rimes, is

tX~XiXt:;:Ti1~

;

but

beeaufe il is poOíble ro hit il every rhrow as well as rhe

lirO, it follows, that the probability of throlVing it

(lnee in four throws, and mifling the olher three, is

~!2i= ~;

which being fubtra{teJ from

rhere

1296 129

will remain

for the probability of throlVing it

once, ano no more, in four times. Therefore, if one

uodemke tOIhrow an aee once, and no more, in fou r

times, he has

500

ro 796 the \Vorí! of rhe lay, Or

ver

oear.

Suppofe tlVO e"enls are fuch, Ihat one of Ihem has

t\Viee as many ehanees ro come up as the olher, what

i! Ihe probabiliry that the eveot, which has Ihe grearer

Dumber of ehanees 10 come up. does nor happeo twice

before tbe orher happeos once, which i! the cafe of

flioging 7 with t\Vo ¿ice before 4 once? Sinee the

oumber of ehances are as

the probabililY of

the fi rn happeoing before the fecond is -r, but the pro·

bability of its happening twice before il is but -rX-r or

therefore it'is 5 tO 4 feveo does not come up Iwice

before four

ooc~.

BUI, if it were demanoed, what mun be the pro·

portion of the facilities of lhe coming up of tlVO e–

vents, to m,ke that whieh has lhe mon ehance! come

np t"ice, before Ihe orher comes up ooce? The ano

flVer is 12,to 5 vcry nearly : whence it follows, that

Ihe prcbability of throlVing lhe fidl before the feeor,d

i;',

aod the probability of throwing it twice is Hx

~~:;

thmfore, the probabiliry of oot doing

it is

~* :

therefore the odd! agaio!! it are as 145 tO

144' IVhich eomts vely ne.r '" equalily,

Suppofe lhere is a heap of thimeo card! of ooe

colour, aod aoother heap of thineen eards of .nother

eolour, what is the probability that. taking ooe eard

a vcnture out of eaeh heap, I ftla ll take out the two

lees?

The prob,bi!it )' of takiog the

ate

out of the firn

heap is·.q. the probability of t,kiog tbe aee out of the

G A M

fee?o~1

he,p i!

,~;

therefore the probabilityof

taking

Out

~oth

aces is

,J'='-}"

whleh being fubt"élcJ

froOl t, Iher< IVdl

rcm.in

~:. :

there(ore the odds

gainll meue 169 to l.

In eafes where rhe events dep,ndononeaoother, the

m,noer of "'guing is fOOlewh,t ahered. Thu!, fup.

pofe thAt Out

one fingle heap ,of Ihine.n eards of

one colour I lhould undcnake to r,ke out

firn.he

m í

anJ, leeondly, the two: thollgh the prob.bili.yofta.

king OUt rhe ace be

Tt,

anll the probabiliry of rakiog

OUI the t

be likewile

,t ,

yet, the ace being fuppo–

led as lak,n out alre,dy, ther.

IV.II

remain only t\VeI,e

eards in the he,p, whlch wi ll make the probability

taking OUt the rlVO tO be ,

~ :

therefNe the rrobability

of taklng OUt the ace, and theo the t.o, will be

TtX

T"t·

In this lan queHion the I\VO eveo!! haveadependellC<:

on each olh", whieh eonliHs in this, Ihat ooe of the

eventl belng luppof-.d 's having happened, the'probabi.

luy of the othds happening is thereby ahereo. Bu!

Ihe caCe is nOt (o in the two heaps of eards.

If the events io quellioo be

in number, and be

fueh as have the fame number

of chaoces by which

th~y

may happen, aod hkewofe the fame number

ehanees by whlch rhey may fail, raife

o+b

10 thepow–

11,

And if A

,lid H

play together, 00 eonJitioo

that if either one or more of the eveots iD quefiioo

happen, A Ihall win, and B lofe, the probabiliryof

A's wiooiog wiH be

~+h)"-b".

aod that of B',

a+b,"

winoiog will be ....!:.. . for wheo

a+h

is a{tually

raifed 10 rhe power

the ooltterm io IVhieh

doe!

not occur is the laH

th~refore

all the terms but

¡he laH are favourable to A.

Tilus,if =3, raifing

o+b

tothe euhe 01+3o

'bt

3Qb'+bl,

al1 the terms but b l wllI be

f.vou~able ~o

A; aod thmfore the probability of A's IVinmng wIII

be~3~¡'+3 1h ', oriF'I_~

aDd the proba.

o+b, l

0+:1

bility of B's wiooing \ViII

be~l'

BUI if Aand B

0+0)

play 00 eondirion, that if either 1\\'0 or mure ,of tbe

eveots in que!!ion happen, A

0,,11

"in; but

c~fe

one only happen, or none, B Ihall IVin; the probablh·

-;c.+,

"-lInb"- '-b" r

ty of A's IVioniog \ViII be '_'"t_DO_I___,___

lor

ntb'l

the only tlVO terms in which

does not occur, arethe

tIVO lall,

viz,

IInO

n- ' and

b".

Gr\MMUT, io mufie, a feale whereoo \Ve Iearo tofound

the mufieal ootes, ./, " ,

I/Ji,jo,fo/,

/11,

iD thm fe·

veral orders aod di(polítioos. Se: MuSIC.

GA NG'WAY il the [everal palTages or ways,

fro~l o~e

part

of the Ihip tO the other

aod whatevcr IS lald

,oy

of thofe p;lT.ges, is faid tO lie io

th~

gang·way ..

GANGEA,' the e.pital of 'territory in

Ih~

provloce of

Chirv,o, in Perlia: E, 100g.

~ 6°,

N, laL 4 IO 'GES

GAN

•