154 AN EXAMINATION OF SOME QUESTIONS 



sing a player adopt this system in betting on a number of in- 

 dependent events, the chance of occurring of each of which is 

 one-half, he would naturally desire to know, if possible, be- 

 fore he began, what would be his profit or loss, supposing p 

 of the events decided in his favour, and q against him. In 

 this particular plan of playing, it so happens, that he cannot, 

 from the mere knowledge of the number of favourable and of 

 unfavourable events, arrive at the conclusion he desires. I 

 shall presently show, that these data alone are insufficient, and 

 that, in order to determine the question, not merely the num- 

 ber, but the order of succession must be given. 



It is probable that this difficulty, in the case of most fre- 

 quent occurrence, has deterred many from attempting other 

 similar problems. Indeed, on the first view of such questions, 

 it is by no means apparent, that any of them can be solved 

 without reference to the order in which the events take place. 

 The mode of inquiry which I shall point out, will show that in 

 many cases their mutual arrangement is not required amongst 

 the data, and will furnish a criterion by which we may deter- 

 mine, in any given case, whether it is necessary. 



I shall first examine the case of the martingal, which, al- 

 though the results it leads to are of a negative nature, will 

 introduce us to the method of treating these questions, and 

 from its frequent practice is rather interesting. 



Let us suppose a gamester bet a certain sum 2 u, upon an 

 event whose chance of happening is one-half; whenever he 

 wins he repeats the same bet ; but whenever he loses, he 

 doubles his last stake. If he should win p and lose q 

 times, it is required to ascertain how much he will have won 

 or lost. 



The first stake being 2w, which may either be gained or 

 lost, we may represent the gamester's profit after one event is 



decided 



