18 ON PROBABILITY. 
(= 1)B, &c. will each == ~ 1, and the remaining n each = — 1, The 
coefficient of @ in the sum of the gains is the sum of all these quantities, 
and therefore =m — n. The coefficient of b is the sum of the products of 
them all two by two, and therefore is equal to the coefficient of the third 
term of an equation which has m roots = + 1, and # roots = — 1, i. e. of 
(x =D" (4 1)", i.e. 
_m.m—1) n.(n=1) _ (m—n)p— (mn) 
Erp = Mack mpept a 3 
— So 2 
Therefore, the player's gain = (m — 2) a + ttn) 5; if 
; ; (m+n) bd Tr 
m = n this reduces itself to Frye, and it is plain that these results are 
quite independent of the order in which the gains and losses follow each 
other. This very elegant solution was given by Mr. Babbage, in the Edinburgh 
Transactions for 1821 ; it remains now to estimate the player's expecta- 
tion, whose probability of winning any single game = p. Let m n= i, 
/ — ae 2 
The player’s gain, if he wins m games, = (m — n) a + ot b 
3 d i(1—1) : : 
= — {iat F520 +emiota- 1)b} —2m.(@m —1) b, since 
n= 1 — m; and in order to get the player’s expectation we must multiply 
this into the term of { p + (1 — p) } of which the argument is p™ . (1-p): =", 
and take the sum of all those products; giving.m, which is now the only 
variable, every value from 0 to ¢ both inclusive. This product is 
Gillme Divi ini Xo iim 
T1.2..m.1.2..aF 1-2; 
po ili=1) ; 
% Yat bt —2m{a4 @—1)b}-+2m(m—1)bl. 
Therefore the sum of all the values of this product is 
: 7.(1—1) Ph She pan ofl Ha Url) 0] wl fii 
fiat 1.2 bf the valuesof .. 1.2..m. 1.2.27 {1=2) 
: : ¢-1.0-2)...1 or 
+2. {ae+( Ddlipatid. covivens 1.2.(m-1.1 2.2" (1-p) 
#500 G2) (fs By.. 1 
- BW l= 1% Xo riers Tetley), 
Rif) pon 1.2.(m-2).1.2.2F U~P) 
When every value from 0 to ¢ inclusive is given to m, the sums of all the 
values of these three right-hand factors, rejecting those in which the index 
of p is negative, severally become {p-(1 —p)}’, {p+ (1 —p)}i-} 
and { p + (1 — p) } ‘= all equal to unity. 
Therefore the sum of all the values of the products 
(~~ 1 : : Si 
= ee) +2ip {a+ GE —-Db}—=2iG — pd 
2.(0—1 
=i.@p-1.a-282 Da, 1) b. 
Ifp = + , this expectation of gain = 2i2a— 2i (i — 1) a®b, and