Prof. Edgeworth's Problems in Probabilities. 179 



several subjects, the candidate's place being determined by 

 the sum of his marks in each ; or (J) there may be only one 

 or two subjects. 



Of the immense number of cases formed by the combina- 

 tions of these attributes I shall discuss only the most inteiest- 

 ing. First in the order of simplicity is Axz. In this case there 

 is a fixed Honour-line, say H ; the comparison is between 

 the place actually obtained by the candidate and the place 

 which he would have obtained if in each subject the marks had 

 been determined by a numerous jury of competent examiners ; 

 the number of competitors may be either many or few ; the 

 number of subjects is large, say S. Since the number of 

 subjects is large and the marking in each subject fluctuates 

 with the change of examiners according to a definite law of 

 frequency, it follows that sums of S marks fluctuate according 

 to the law of error, the Probability Curve. Let C be the 

 Mean Deviation for the marking in each subject ; in the 

 sense above defined, that is \/2 x the Gaussian Mean Error. 

 Then the Probability-curve, according to which the compound 

 mark determined by any set of S examiners will fluctuate, 

 has for its Modulus the coefficient C ; which is ascertainable 

 by observation. Xow suppose the candidate has obtained the 

 mark H + /. The problem may be likened to the familiar one: 

 If the average of S observations of given precision be H + /, 

 what is the probability that the true value is less than H. By 

 received reasoning the probability in question is 



■wCJi 



V 



which may be calculated from the usual tables. 



Ksez. Tins case differs from the preceding in that the 

 comparison of the actual compound mark is not with the true 

 mark, but with the mark winch any other set of S competent 

 examiners might have assigned. It is as if, in the parallel 

 problem, we sought the probability, not that the true value 

 is less than H, but that any second measurement made 

 under similar conditions should fall below H. According to 

 a well-known theory the solution is obtained by substituting 

 V2 x C for C in the solution of the preceding problem. 



Bj'i/z. This is the case in which a fixed large number, say >*, 

 prizes are assigned to the n candidates who come out first 

 (irrespectively of the absolute number of their marks); and 

 the inquiry is whether any particular candidate would have 

 his status changed from successful to unsuccessful, or vice 



N2 



