288 d'alembert. 



531. D'Alembert then proposes a method of solving the Peters- 

 hiirg Prohlem which shall avoid the infinite result ; this method is 

 perfectly arbitrary. He says, if tail has arrived at the first throw, 



let the chance that head arrives at the next be ^ , and not 



2 



- , where a is some small quantity ; if tail has arrived at the first 

 throw, and at the second, let the chance that head arrives at the 



next throw be ^ , and not ^ ; if tail has arrived at the first 



throw, at the second, and at the third, let the chance that head 



arrives at the next throw be — , and not - ; and so on. 



^ Ji 



The quantities a, h, c, ... are supposed small positive quantities, 

 and subjected to the limitation that their sum is less than unity, 

 so that every chance may be less than unity. 



On this supposition if the game be as it is described in Art. 389, 

 it may be shewn that A ought to give half of the following series : 



1 



+ (!+«) 



-f (1 - a) (1 + a + ^) 



+ (1 - «) (1 - a - Z>) (1 + a + Z> + c) 



■^(l-a){l- a-h-c) {l + a-\-h + c-\-d) 



+ 



It is easily shewn that this is finite. For 



(1) Each of the factors 1+a, \ -\-a-\-h, l + a + Z>+c, ...is less 

 than 2. 



(2) \ — a — h is less than 1 — a\ 



1 — a — 5 — c is less than 1 — a — h, and a fortiori less than 

 1 — a ; 



and so on. 



Thus the series excluding the first two terms is less than the 

 Geometrical Progression 



2 {1 - a + (1 - a)^ -f (1 - ay + (1 _ a)\ . .), 

 and is therefore finite. 



