540 LAPLACE. 



event will happen i times in succession. Then from equation (1) 

 of Art. 678 by changing the notation we have 



^ (x) ^p'+p'-' (1 -p) cj, (x-i) -\-p'-' (1 -p) ct>{x-i-^ 1) + ... 



,..+p{l-p)c^{x-2) + (l^p)<t>(x-l) (1). 



Laplace takes z^ to denote the probability that the run will 

 finish at the x^^ trial, and not before ; then he obtains 



^x = (1 p) 1^.-1 + P^^, +P\-, + . . . + p"' ^x- j (2). 



"VVe may deduce (2) thus ; it is obvious that 



z^ = (f>{x)-(f){x-l); 



hence in (1) change x into x — 1 and subtract, and we ob- 

 tain (2). 



Laplace proceeds nearly thus. If the run is first completed 

 at the x^^ trial the (x—iy^ trial must have been unfavourable, and 

 the following i trials favourable. Laplace then makes ^ distinct 

 cases. 



I. The (x — i— \y^ trial unfavourable. 



IL The {x-i-iy^ favourable; and the (a;-f-2y^ un- 

 favourable. 



Ill The [x-i- \y^ and the {x - i- 2)"^ favourable, and the 

 {x — {— 3) ''^ unfavou rable. 



IV. The {^x-i-Vf\ the {x-i-2f\ and the {x-i-Zf 

 favourable; and the (a;— ^ — 4)^^ unfavourable. 



And so on. 



Let us take one of these cases, say IV. Let P^ denote the 

 probability of this case existing ; then will 



For in this case a run of 3 has been obtained, and if this be 

 followed by a run of /— 3, of which the chance is p^'"^, we obtain 

 a run of i ending at the [x — 4)*^ trial. 



Now the part of z^ which arises from this case IV. is FJ(\. —p) j/\ 

 for we require an unfavourable result at the {x — if^ trial, of 



