CONNECTED WITH GAMES OF CHANCE. T55 



decided by 2m ( — l) a , a being any whole number; for since 

 the nature of the number a is left undecided, whether it is an 

 even or an odd one, the expression just given will represent 

 either a profit or a loss. First, if a is an even number, his 

 next stake remains the same as it was before, or equal to 2 w, 

 if it is an odd number, his second stake must be 4 u. We must 

 therefore endeavour to find some function of a which shall be 

 equal to 2w, when a is an even number, and hecome 4 m, 

 when a is an odd one. A great variety of functions may be 



a 

 1 — (-D 



found satisfying this condition ; such are 2 u X 2 



2 u ( * c T a j 9 &c. and many others. Our choice amongst 



the infinite variety which present themselves, must be direc- 

 ted by the ulterior operations of the gamester ; and, first, I re- 

 mark, that every stake must be equal to the constant quantity 

 w, multiplied by some power of 2. The next remark which 

 may guide us in this choice of a proper function is, that the 

 index of that power of 2 is determined by the number of times 

 (in immediate succession) the event has been unfavourable, 

 reckoning back from the event about which the stake is pro- 

 posed : if, therefore, at any period, all the preceding determi- 

 nations have been unfavourable, the amount of the next stake 

 will be influenced by them all. It may also be noticed, that 

 the successive powers of 2 can be formed by the addition of 

 all the preceding ones, together with unity, thus, 



1, 2°, 2 1 , 2% 2% 2% 2 5 

 l + 2°=2, l+2° + 2 1 =r4, 1 -f 2° + 2 1 + 2 2 = 8, and so on : 



From these considerations it appears, that the function we re- 

 quire may consist of a series of functions, each multiplying the 



u 2 successive 



