ON PROBABILITY. 17 
solutions to the problem preceding, and similar problems analogous to those 
obtained on the supposition of Buffon. 
We shall nos dwell longer on this subject, ‘as these hypotheses, although 
they may serve, in some measure, to show the difference which exists 
between the mathematical value of any sum and its value in practice, are 
quite arbitrary. 
aS) 26. The search after a method of enabling a gamester to win with cer- 
pas tainty from his antagonist, who has a greater probability in his favour, has 
et than wasted as much ingenuity as the attempt to discover perpetual motion. At 
ut, "Ig all gaming tables an advantage is given by the laws of the game to the 
ta banker ; and many infatuated persons, in the vain hope of detecting some 
scheme for rendering that advantage nugatory, have spent years in register- 
De ing the course of the play with a degree of patient industry which, exerted 
: in another direction, might have made them useful and distinguished 
4: members of society. One favourite scheme is so celebrated as to have 
el acquired a particular name; it is called the Martingale, or Double or Quits, 
and consists in doubling the last stake after every loss. In order that this 
may be permanently successful, the player requires not only an immense 
Lh capital, but an unlimited permission of staking. It is not very easy to show 
mathematically the amount of the player's expectation who uses the martin- 
gale, on account’ of the various order in which the gains and losses may be 
a supposed to follow each other; instead of attempting it, we shall give an 
a analysis of another scheme, in which the same difficulty does not occur. 
This consists in increasing the stake by a fixed sum after every loss, and 
diminishing it by the same after every gain. The inventors of this mode of 
betting looked upon it as infallible, and indeed there is something in it 
which might easily deceive the unwary ; for it can be shown, that if the 
+ 0 number of games won and lost be the same, no matter in what order this 
: takes place, the result is always a gain to the player who bets upon this 
therefore, principle. Notwithstanding this specious circumstance, we shall show that 
f+na), the value of the player's expectation of gain, when his probability of winning 
a single game exceeds %, is never so great as his expectation of loss when 
this probability falls short of Z by the same quantity. 
27. Let a be the original stake, 6 the quantity by which it is increased or 
id 8 diminished after every loss or gain, m + n the whole number of games 
played. The first thing to determine is the player’s gain who wins m and 
ame vessel, loses n games. 
¢ merchant, His first stake = a, therefore his first gain = + «, and may be represented 
 Tison generally by @ (— 1)# which becomes + aas ais even orodd. Ifhe wins, 
Fo 2. e. if abe even, his second stake is @ — b, and it is @ + b if he loses, i. e. if 
VF eaested a be odd: therefore his second stake generally is @ — b (— 1)# and his 
wi second gain may be represented in the same general manner by{a—25 
Fay ind (—1)#} (= 1)B. For the same reason his third stake will be & — & (= = 
elie — b. (= 1)A, and therefore his third gain is { @ — b ((— 1) + (~ 1)B)} 
pr (= 1)% and so on. 
Re eee? We have, therefore, the following table of gains : 
: Ist gain = a (= 1), 
2d. =a (~DB=b{(=1)2} (=1)8, 
8d .. =a(—1)r—b{(— 1) (=1)8 } (=1)¥, 
4th... =qg(— 1) — b{ (= 1)z-f(—1)8 +(-D7} (=D), 
The dsr &e. &e. &e. i; 
te 2’ There are m + n stakes, and we have determined nothing of the quantities 
oly UF a, 3, 7, &c. except that m are even, and 7 odd, because by supposition the 
fonds player wins m_ games, and loses n, Therefore m of the quantities (— 1),