LAPLACE. 593 



proximation take the integral between the limits and 





2 /* 



and double the result : he says this amounts to neglecting the 

 square of t'^ — rl We may put the matter in the following form : 

 suppose that a and h are positive, and we require x such that^ 



/. 



"'e-''dt+{\-^dt = 2\ ""e-^' dt 



Suppose a less than h ; then in fact we require that 



a J X 



'a' + 1' 



Laplace, in effect, tells us that we should take x = 



2 ; 



as an approximation. He gives no reason however, and the more 



natural approximation would be to take x = ^ (a + 5), and this is 



certainly a better approximation than his. For since the function 

 e~^^ decreases as t increases, the true value of x is less than 



1 . , . . . 1 



^ (a + 5), while Laplace's approximation is greater than ^ (a + V). 



1024. Laplace discusses on his pages 869 — 376 a problem re- 

 lating to play ; see Art. 868. A and B play a certain number of 

 matches ; to gain a match a player must win two games out of 

 three ; having given that A has gained i matches out of a large 

 number n, determine the probability that -4's skill lies within as- 

 signed limits. If a player wins the first and second games of a 

 match the third is not played, being unnecessary ; hence if n 

 matches have been played the number of games must lie between 

 2/i and 3?i : Laplace investigates the most probable number of 

 games. 



1025. Laplace discusses in his pages 377 — 380 the problem 

 which we have enunciated in Art. 896. The required proba- 

 bility is 



x'i^-xYdx 



i: 



[ x^ii-xydx 



■^ 



where p and q have the values derived from observations during 



38 



