Encyclopaedia Britannica; or, A dictionary of arts and sciences, compiled upon a new plan … > Volume 2, C-L

GAM
deration. Thus if I would know what the probability
is of miffing an ace four tint s together with a die, this
I confider as the failing of four different events. Now
the probability of miffing the firft is the fecond is
alfo ■§-, the third and the" fourth therefore the
probability of miffing it four times together is
X4=-rrf|-; which being fob traded from i, there will
remain -rf^ for^the probability of throwing it once or
oftener in.four times: therefore the odds of throwing
an ace in four times, is 671 to 625.
But if the flinging of an ace was undertaken in three
times, the probability of miffing it three times would
be gX|-X{-—4 r'o ’ which being fubtraded from 1, theie
will remain ^or the probability of throwing it once
or oftener in three times : therefore the odds againlt
throwing it in three times are 12 J to 91. Again, fup-
pofe we would know the probability of throwing an ace
once in four times, and no more : fince the probability of
throwing it the firft time is •§■, and of miffing it the o-
ther three times is ^X£X£, it follows that the proba¬
bility of throwing it the firft time, and miffing it the
other three fucceffive times, is •6X^X|-X|-=T1^J ; but
becaufe it is poffible to hit it every throw as well as the
firft, it follows, that the probability of throwing it
once in four throws, and miffing the other three, is
4^1?!=;-£2?; which being fubtraded from 1, there
1296 1296
will remain -r-rlf ^or l^e probability of throwing it
once, and no more, in four times. Therefore, if one
undertake to throw an ace once, and no more, in four
times, he has 500 to 796 the worft of the lay, or 5 to
8 very near.
Suppofe two events are fuch, that one of them has
twice as many chances to come up as the other, what
is the probability that the event, which has the greater
number of chances to come up, does not happen twice
before the other happens once, which is the cale of
flinging 7 with two dice before 4 once ? Since the
number of chances are as 2 to 1, the probability of
the firft happening before the fecond is y, but the pro¬
bability of its happening twice before it is but ^X-f- or
£: therefore it is 5 to 4 feven does not come up twice
before four once.
But, if it were demanded, what muft be the pro¬
portion of the facilities of the coming up of two e-
vents, to make that which has the moft chances come
up twice, before the other comes up once ? The an-
fwer is 12 to 5 very nearly: whence it follows, that
the probability of throwing the firft before the fecond
is 4-4, and the probability of throwing it twice is 44X
44, or ^41therefore, the probability of not doing
it is f4|-: therefore the odds againft it are as 145 to
144, which comes very near an equality.
Suppofe there is a heap of thirteen cards of one
colour, and another heap of thirteen cards of another
colour, what is the probability that, taking one card
at a venture out of each heap, I fhall take out the two
aces ?
The probability of taking the ace out of the firft
heap is -ry, the probability of taking the ace out of the
G A M
fecond heap is 74 ; therefore the probability of taking
out both aces is T4XT4=T49, which being fubtraded
from 1, there will remain 4-f-y-: therefore the odds a-
gainft me are i63 to 1.
In cafes where the events depend on one another, the
manner of arguing is fomewhat altered. Thus, fup-
pofe that out of one Angle heap of thirteen cards of
one colour I fhould undertake to take out firft the ace;
and, fecondly, the two : though the probability of ta¬
king out the acebeT4> and the probability of taking
out the two be likewife yf; yet, the ace being fuppo-
fed as taken out already, there will remain only twelve
cards in the heap, which will make the probability of
taking out the two to be Ty; therefore the probability
of taking out the ace, and then the t»o, will be TfX
-ri-
In' this laft queftion the two events have a dependence
on each other, which confifts in this, that one of the
events being fuppofed as having happened, the probabi¬
lity of the other’s happening is thereby altered. But
the cafe is not fo in the two heaps of cards.
If the events in queftion be « in number, and be
fuch as have the fame number a of chances by which
they may happen, and likewife the fame number b of
chances by which they may fail, raife a-\-b to the pow¬
er n. And if A and B play together, on condition
that if either one or more of the events in queftion
happen, A fliall win, and B lofe, the probability of
A’s winning will he kn . and that of B’s
a+b?
winning will be —t— ; for. when a-\-b is adually
raifed to the powers, the only term in which a does
not occur is the laft bn : therefore all the terms but
the laft are favourable to A.
Thus if »=3, raffing a-\-b to the cube
3,ab'l-\-bl, all the terms but b* will be favourable to
A ; and therefore the probability of A’s winning will
be— nra+^ ~^3 . and the proba-
aJ\rb)3 a-jpiTl5
bility of B’s winning will be—t - . But if A and B
play on condition, that if either two or more of the
events in queftion happen, A ftiall win; but in cafe
one only happen, or none, B ftiall win; the probabili¬
ty of A’s winning will be^~^... ^ ; for
w-HV1
the only two terms in which aa does not,occur, are the
two laft, viz, nab0-*-1 and ba-.
GAMMUT. in mufic, a fcale. whereon we learn to found ;
the mufical notes, ut, re, mi, fa, fol* la, in their fe-
veral orders and difpofitions. See Music.
Gang-way is the feveral paffages or ways from one part
of the (hip to the other ; and whatever is laid in any
of thofe paffages, is faid to lie in the gang-way,
GANGEA, the capital of a territory in the province of "
Chirvan,. in Perfia: E.long. 46°, N. lat. 410.
GANGES,
( 644 )