182 Prof. Edgeworth's Problems in Probabilities. 



II, The answers which have been given to the first problem 

 are required for the solution of the second problem : At any 

 examination of which the circumstances are given, how many 

 of the candidates are uncertain in this sense, that there is an 

 appreciable chance of any assigned one of them who is now 

 successful coming out unsuccessful, and vice versa ; if the 

 work were marked by different but equally competent ex- 

 aminers? If we confine ourselves to the general case of 

 several candidates and several subjects (yz), we have only to 

 measure from the Honour-line in both directions a distance 

 such that the probability of any candidate at this distance 

 being displaced is very small, say less than one in a hundred. 

 This improbable error, or discrepancy as it may be called, is 

 found by multiplying the probable error, or discrepancy, 

 proper to the case by 3*5. The candidates above that limit 

 may be described as " safe." 



The reader who applies this formula to statistics of exami- 

 nations, such as those which are given in the Reports of the 

 Civil Service Commissioners, may be surprised to find that 

 the number of the uncertain unsuccessful is greater than that 

 of the uncertain successful ; although, in the case of a deter- 

 minate number of prizes (B), every instance of a successful 

 candidate being in the wrong box involves an instance of an 

 unsuccessful candidate being misplaced. The explanation of 

 this anomaly is that, in applying the received formula, we 

 have made the common assumption that the a priori proba- 

 bility of the candidate's real mark, so to speak, being one 

 figure rather than another is constant. The nature of that 

 assumption and the caution with which it must be made* are 

 well illustrated by these problems. In the present case the 

 a priori probabilities are not constant. In general the marks 

 of candidates at an examination are not distributed equably 

 between the positions of the senior and the man at the bottom; 

 but are heaped up in the form of a Probability-curve f. Now 

 the scene of our operations is the upper extremity of this 

 Probability-curve ; whence it follows that the a, priori pro- 

 bability (for each point or degree) diminishes as we ascend 



* See my paper "On A priori Probabilities " in the Philosophical 

 Magazine for September 1884 ; also ' Metretike ' (London, Temple Co., 

 1887). 



t With respect to this statement and others which may seem to 

 require proof the reader is again referred to companion papers in the 

 Journal of the Statistical Society, Sept. 1888 and Sept. 1890. 1 have 

 sometimes in those papers used the term "true mark" for the mean of 

 the marks given by an indefinite number of examiners — a conception which 

 is not absolutely essential to the x variety of our problems. 



