ON PROBABILITY. 
TN tame ] — gt! 
a? 42! ali. ee Haat mpm 
f : i.(i-1) i.(-1).(G~2) 
—t Py i pT! Zep), FN TSN LEERY) % 
{1 : } Ir i 1.2.3 aa 1 é& 
Lt. 1) t.(eF1.G+ 2) 
1 a— m= — 1 dl 2 NT eay———— 3 
{ =} 7 1.2.3 2% 
The coefficient of 2° is obtained by multiplying the coefficient of the first 
term of the upper “series by the coefficient of 2° in the lower, that of the 
second in the upper by that of 2° =! in the lower, &e. and 
_t.(+1) E42)... (E+s—1) Es C+1D.GC+2)...G + s—n—2) 
Ee 1.2.3....s(n+ 1) 1.2.3....6—n—=1) (n +1)" 
g t.¢—1D1. G+ D.C+2)....0+5~-2n— 3) : 
rat A wn me ese ere i 
+ 1.2 1.2.3.... s—2n—2) (2 +4 1) &e 
This series is equivalent to 
GCH+D.6+2)..s+i—=1) ; (s—n).(s =n D..6=n+i-92) 
io. aN TTT tyre TY 
23 mwill be 1.200 vn E—1) (n+1) 2 ST 
NL 1) €—22~ 1) (s+~2p)....6~2n4i~— 3) 
. . 
1.2 13 UDOT 2 
It is to be remarked, that all the terms of the developement of (1 — ar hiy 
in which 2 is involved to a higher power than s may be rejected, because we 
have no negative powers of x in the other factor (1 — 2) -* by which to 
A will win m reduce them. 
at A will win This serves to show how many terms of the resulting series are to be 
and. there- taken, for if / represent the rank of the last term, we must have ¢-1 
srobabilties, (n +41) = <5. Therefore the series is to be continued only so long as 
s of the form s— (lL —1).(n + 1) is positive. As a numerical example, we may take 
sented bv the the problem already solved in page 4, to find the chance of throwing 7 with 
oliet shape, two dice. Dice have no side marked with 0, therefore, before the formula 
Therefore the can be applied to this and similar cases, each face or ball must be supposed 
te Bret erm to import one less than is marked on it, which amounts to substituting in 
the wilt the formula s — 7» for s. We shall, however, not substitute in the formula 
oo so altered, but deduce its value in the particular case, as we have done in 
a 6 the general one. 
ons The probability required will be the coefficient of a7 in 
- x + a + 2° + Ade r—a’ 
sobablities ! 6 B= Ss 
oroposed {0 1 
os wilh ice. = (2 =2a" +2") 1 +22 +32" + 44° 5a" 4 62° + &.) 
the numb 6 1 
? how. which is HE before. 
If we want the probability of throwing 10 with three dice, we must find 
the coefficient of #'° in 
ol fete ay f z—a F 
geanse 1 en de 
6 } 16.01 — 4) 
1h diff I 1 . 
it = Ta (@=32" + &e) (1 + 32-622... +3627 + &e.) 
13