in Mathematical Reasoning. 341 
and the profit or loss by that bet will be 
DGS OE ak 4 ale 
a new undetermined letter being again introduced. 
By these means his xth stake will be represented by 
th hea Cra aie Ga Mee ee LS mais. 1) 
and thus all the x cases become reduced into one expression. 
The influence of this indeterminateness in contributing to the 
success of the solution of the following problem is very apparent. 
Supposing a player to regulate his stakes in the manner just 
described, and that his chances of winning and losing are equal, 
what will he have profited after p+q events, p of which have been 
favorable, and q have been unfavorable ? 
At a first view it might perhaps be considered that the. data 
are insufficient for the solution, and that the order in which the 
events take place ought to be given: the result however of the 
investigation will prove that the profit or loss is entirely indepen- 
dent of that consideration. 
It has already been shewn that his profit on each event will 
be as follows: 
u(-1)', 
ui = 1Y—O4 (A Cady 
w (Hh) pen (=) Hol Se 2S 
Mal) ay Wieie yi ae 1)°$ ae 
u(—1)" =» {(—3)" +(<1)+ (ou + (-0). 
On examining the composition of that collection of terms which 
multiply v it will appear that the sum of all the above quantities 
is equal to 
uf(—1l + (-1f +...(—1)}, 
<x fine sum of all the site two by two which aa 
be formed out of the quantities (- 1)’, (—1)’,...(—1) 
Sey Z 
