APPENDIX THE FIRST. 



stakes, play at a game which gives a to & in favour of A 

 at each trial, then,, in an indefinite number of trials 



the chance that B sha11 ruin 



A the chance that A shall ruin B 



nf 7 m m *\ 

 a \^ b a ) 



vt MHSUiiailvc uiai, -n. simii i uiii j 



, n, ,n + m n + m 



b a 



The sum of these two chances is unity, from which it 

 appears that one or other must be ruined in the long run. 

 These results agree entirely with those of De Moivre, 

 and all the rules in page 109. may he easily deduced 

 from them. 



It appears also, that if the conditions of any game, 

 however complicated, can be reduced, in the case where 

 one of the players (A) has unlimited means, to the 

 equation 



then the ultimate results of that game exhibit probabi- 

 lities of precisely the same value as those of a simple 

 game, in which it is a to /3 for A against B. 



If, besides cases in which A or B wins, there be 

 cases in which the game is drawn, no alteration is made 

 in the result (though the number of games in which 

 there is a given probability of either party winning 

 must of course be increased.) Let a, /3, and , be the 

 chances that A wins, that B wins, and that the game is 

 drawn: then (a + j3 + 8=1) 



B,OQ = B w + i }00 -j-a B w _ i )00 _j_5 B w>00 or 



a / j8 a \ 



B == iTT5 Bw + 1 co +rT5 Bw - 1 ' co Vn5 + TH8 = V 



the same as in a game which must be won or lost, and 

 in which it is a to /3 for A against B. 



The following is the problem of the game of rouge et 

 noir, which I shall afterwards proceed to explain. 



