CONDORCET. 393 



142 — 147. The sum of the remarks is this ; Condorcet justifies the 

 rule on the ground that it will lead to satisfactory results if a very 

 large number of trials be made. Suppose for example that A and 

 B are playing together, and that -4's chance of winning a single 

 game is p, and 5's chance is q : then the rule prescribes that if -4's 

 stake be denoted by hp, then ^'s stake must be hq. Now we 

 know, by Bernoulli's Theorem, that if A and B play a very large 

 number of games, there is a very high probability that the number 

 which A wins will bear to the number which B wins a ratio ex- 

 tremely near to the ratio oi p to q. Thus if the stakes are adjusted 

 according to the general rule there is a very high probability that 

 A and B are on terms of equality as to their prospects ; if any 

 other ratio of the stakes be adopted a proportional advantage is 

 given to one of the players. 



There can be no doubt that this view of the ground on which 

 the rule is to be justified is correct. 



723. Condorcet adverts to the Petersburg Problem. The 

 nature of his remarks may be anticipated. Suppose that p in 

 the preceding Article is extremely small and q very nearly equal to 

 unity. Then ^'s stake is very large indeed compared with ^'s. 

 Hence it may be very imprudent for B to play with A on such 

 terms, because B may be ruined in a few games. Still it remains 

 true that if A and B agree to continue playing through a very 

 long series of games no proportion of stakes can be fair except that 

 which the general rule assigns. 



724. The second part of Condorcet's memoir is entitled Ap- 

 plication de r analyse a cette question: Determiner la probabilite 

 quun arrangement regulier est Veffet d'une intention de le pro- 

 duire. 



This question is analogous to one discussed by Daniel Ber- 

 noulli, and to one discussed by Michell ; see Arts. 395 and 618. 



Condorcet's investigations rest on such arbitrary hypotheses 

 that little value can be attached to them. We will give one 

 specimen. 



Consider the following two series : 



1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 



1, 3, 2, 1, 7, 13, 23, 44, 87, 167. 



