554i LAPLACE. 



Suppose that the probability of the birth of a boy to that of 

 the birth of a girl be as 18 to 17 : required the probability that 

 in 14000 births the number of boys will fall between 7363 and 

 7037. 



Here 



i?=^, q = ^, m=7200, n = 6S00, r=163: 



the required probability is '994303. 



The details of the calculation will be found in Art. 74 of the 

 TJieory of Prohdbilities in the EncyclopcEdia MetropoUtana, 



997. We have now to notice a certain inverse application 

 which Laplace makes of James Bernoulli's theorem : this is a 

 point of considerable importance to which we have already alluded 

 in Art. 125, and which we must now carefully discuss. 



In Art. 993 it is supposed that p is given, and we find the 

 probability that the ratio of the number of times in which the 

 event happens to the whole number of trials will lie between 

 assigned limits. Suppose however that p is not known a pi^oriy 

 but that we have observed the event to happen m times and to 

 fail n times in fi trials. Then Laplace assumes that the expression 



given in Art. 993 will be the probability that p— — lies be- 



ft 



tween 



• ^^/(2mn) ^^^ ^ T>sJ{2mn) ^ 



that is, Laplace takes for this probability the expression 



i-\\-i^di^ ,,y^ ,6-- (1). 



He draws an inference from the formula, and then says, on 

 his page 282, 



On parvient directement ^ ces resultats, en considerant p comme 

 une variable qui pent s'etendre depuis zero jusqu'a I'unite, et en deter- 

 minant, d'apres les evenemens observes, la probabilite de ses diverses 

 valeurs, comme on le verra lorsqne nous traiterons de la probabiUte des 

 causes, deduite des evenemens observes. 



