LAPLACE. . 471 



If we suppose that tis may have any value between and c we 

 may multiply the last expression by d-us and integrate from to c. 

 See Art. 529. 



878. As another example Laplace considers the following 



question. A undertakes to throw a given face with a common die 



in n throws : required his chance. 



/5\" 

 If the die be perfectly symmetrical the chance is 1 — i -j ; but 



if the die be not perfectly symmetrical this result must be 

 modified. Laplace gives the investigation : the principle is the 

 same as in another example which Laplace also gives, and to which 

 we will confine ourselves. Instead of a common die with six faces 

 we will suppose a triangular prism which can only fall on one of its 

 three rectangular faces : required the probability that in n throws 

 it will fall on an assigned face. Let the chance of its falling on the 



three faces be — ^ — , — ^ — and — - — respectively, so that 



'UJ ■\- "UJ ■\- 'US =0. 



Then if we are quite ignorant which of the three chances belongs 

 to the assigned face, we must suppose in succession that each of 

 them does, and take one-third of the sum of the results. Thus we 

 obtain one-third of the following sum, 



{ 



If we reject powers of ■or, tzr', and -ot" beyond the square we get 

 approximately 



«?« n (n _ I) 9"-2 



3'' 1 2 ' • 3'^ '^ +tu- +-57 ;. 



Suppose we know nothing about ct, ot', and ot", except that 

 each must lie between — c and + c ; we wish to find what we may 

 call the average value of ot^ -|- tn-'^ -h ts-''^ 



We may suppose that we require the mean value of x^ + 1^ -|- z^, 



