APPENDIX. 



APPENDIX THE FIRST. 



ON THE ULTIMATE CHANCES OF GAIN OB LOSS AT PLAY, 

 WITH A PARTICULAR APPLICATION TO THE GAME 

 OF ROUGE ET NOIR. 



THOUGH the first part of the following reasoning is 

 of a mathematical character, I have been induced to 

 insert it by the consideration that the results of page 109. 

 have never yet been introduced into an elementary 

 work, nor even proved to the mathematician except either 

 by incomplete or complicated trains of reasoning. Such 

 being the case, perhaps even a well-informed mathe- 

 matician might be excused for doubting some of the 

 results of chapter V., and I have therefore digested the 

 following demonstration, that no one who bears such a 

 character may be able to weaken the evidence for the 

 necessity of the pernicious results of gambling which 

 that chapter is intended to afford. 



De Moivre was the first who gave a solution of the 

 following problem, and by a method of the most striking 

 ingenuity. But his demonstration has the defect of 

 assuming that one or other of the players must be 

 ruined in the long run. Laplace* and Ampere, the 



* The solution of Laplace gives results for the most part in precisely 

 the same form as those of De Moivre, but, according to Laplace's usual 

 custom, no predecessor is mentioned. Though generally aware that La- 

 place (and too many others, particularly among French writers) was much 

 given to this unworthy species of suppression, I had not any idea of the 

 extent to which it was carried until I compared his solution of the problem 

 of the duration of play, with that of De Moivre. Having been instru- 



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