ULTIMATE RESULTS OF PLAY. Xlll 



or V - (A) OT V - ( or 



from which it is readily found that 



. AV, - * ( V, *-4AV x _ 10 _V ;c . 9 . ) 



The first eleven values of V x are thus determined : V t is 

 evidently ^; V 2 is ^ V t + ^ or & . fa V 8 is T ' 3 Vj + -^ V 2 

 + T3 or A (la) 2 ; anc ^ so on U P to ^10 wa ich is 



V n is T 3 ( Vio + V 9 . . . + V 2 ) 



The rest were then calculated by the preceding for- 

 mula for AV.J. to six places of decimals (by which the 

 accuracy of four was insured) as far as Vsi inclusive. 

 The remainder were calculated by those which preceded, 

 leaving out the terms which the necessary distinction 

 already mentioned requires to be omitted. 



It may perhaps appear to some that a part of the 

 preceding reasoning is inapplicable, since it only calcu- 

 lates the chances of ruin in an indefinite succession of 

 gam es_, whereas any practicable number of games, though 

 great, may not involve the same chances as an infinite 

 number. The objection is valid in principle, but the 

 correction which is rendered necessary by it is not worth 

 consideration, if any large number of games be in ques- 

 tion. 



The following are the only cases in which a simple 

 approximate rule can be given, connected with finite 

 numbers of games : 



PROBLEM. Both parties have the same number of 

 stakes (which should not be less than 20), say a, and 

 the play is equal, or either has an even chance of win- 

 ning any one game. What is the chance that one or 

 other shall have been ruined before a? games have been 

 played ? (oc being a large number). 



