208 SIMPSON. 



363. Simpson's Problem XX. is the same as De Moivre's Pro- 

 blem VII ; it is an example of the Duration of Play : see Art. 107 ; 

 Simpson's method is less artificial than that which De Moivre used, 

 and in fact much resembles the modern method. 



364. Simpson's Problem xxil. is that which we have explained 

 in Art. 148 ; Simpson's method is very laborious compared with 

 De Moivre's. Simpson however adds a useful Corollary. 



By introducing or cancelling common factors we may put the 

 result of Art. 148 in the following form : 



(p'-l){p-2) ... (p-n-\-l) _n fe-1) fa- 2). ..(^-72+1) 

 \n — l 1 1^ — 1 



n{n — l) (r— 1) (r — 2) ... (r — n + 1) 



where g^—'p-f, r — 'p—^f, ...; and the series is to continue so 

 long as no negative factors appear. 



Simpson's Corollary then assigns the chance that the sum of the 

 numbers exhibited by the dice shall not exceed p. We must put 

 successively 1, 2, 3, ... up to p for p in the preceding expression, 

 and sum the results. This gives, by an elementary proposition 

 respecting the summation of series, the following expression for the 

 required chance : 



p{p-l) ,..{p — n-\-l) n q{q-V) ... (q-n + l) 

 [n 1 [n 



n(n—l) r (r—1) ... (r — n + 1) 



where, as before, the series is to continue so long as no negative 

 factor appears. 



365. Simpson's Problem xxiv. is the same as De Moivre's 

 LXXiv., namely respecting the chance of a run of p successes in 

 n trials ; see Art. 325. De Moivre gave the solution without a 

 demonstration ; Simpson gives an imperfect demonstration, for 

 having proceeded some way he says that the '' Law of Continuation 

 is manifest." 



