470 LAPLACE. 



a certain curve of probability bisects the area of the curve. See 

 Theorie...des Froh. page 335. 



Laplace does not notice another sense of the word mean, 

 namely an average of all the values ; in this case the mean would 

 correspond to the abscissa of the centre of gravity of the area of 

 a certain curve. See Art. 485. 



877. Laplace now proceeds to the subject which is considered 

 in Chapter VIL of the Theorie...des Froh., namely the influence 

 produced by the want of perfect symmetry in coins or dice on the 

 chances of repetitions of events. The present memoir and the 

 Chapter in the Theorie...des Froh. give different illustrations of 

 the subject. 



The first case in the memoir is that of the Fetershurg Fro- 

 hlem, though Laplace does not give it any name. Suppose the 



chance for head to be — ^ — , and therefore the chance for tail 



to be — ^^ — ; suppose there are to be x trials, and that 2 crowns 



are to be received if head appears at the first trial, 4 crowns if 

 head does not appear until the second trial, and so on. Then the 

 expectation is 



If the chance for head is — ^ — , and therefore the chance for 



tail is —^~ , we must change the sign of in- in the expression for 



the expectation. If we do not know which is the more likelv to 

 appear, head or tail, we may take half the sum of the two expres- 

 sions for the expectation. This gives 



If we expand, and reject powers of ot higher than -cj^, we obtain 



