the Theory of Probabilities -' ^ '' "^ ^ 57 



- ' The proposed problem, therefore, as applying to ordinary ca?es, 

 has never been, and at present cannot be, solvedi'*'*^ Utj^ji yiUiifJ 

 But it is to be observed, that if it were solved, that is^ if the^' 

 summation just mentioned were actually performed, it cannot be 

 assumed beforehand that the result would not turn out to be of 

 the forai Q'^{x — a)'^{oo — a!).,.,dx, giving the same relative 

 probabilitij for any value of x as would be obtained if it were 

 known that •\/r(^) were the function actually expressing the law of 

 facility of errors in the individual obse7^ations. Such a result 

 would involve no prima facie absurdity or difficulty, and it would 

 not be a valid objection to it to say, that it professed to establish 

 an independent external reality by a priori mathematical rea- 

 soning. For to prove that a required probability is to be calcu- 

 lated as if a certain hypothesis were known to be true, is a per- 

 fectly different thing from proving that that hypothesis is true, 

 or from proving anything about the probability of its truth at 

 all. To take a simple analogous case, suppose a bag contains 

 an unknown number of balls, of unknown colours ; a ball is 

 drawn and replaced n times, and is white each time ; now if a 

 person professes to prove that the probability of di-awing a white 



ball at the next trial is , we may object to his proof on oth0? 



grounds, but certainly not on the ground that he thereby assumes 

 this to be the actual ratio of the number of white balls to the 

 whole number of balls. Of course his answer would be, that he 

 assumes no such thing, but only asserts that the probability rela- 

 tive to a certain state of information is the same as it would be 

 if a certain hypothesis were known to be true. The fallacy con- 

 sists in assuming, that because two probabilities are equal, the 

 states of information to which they refer must be identical. • 

 To return to the subject of observations. If the law of facility 

 of errors were known, the mean of the observed values would not 

 be the most probable result, unless the law were expressed by 

 the well-known exponential function assigned by Gauss in his 

 first investigation. But the law of facility not being known, 

 although it has never been proved that the mean is the most 

 probable result, relative to this state of information, it has cer- 

 tainly never been proved that it is not : the question is perfectly 

 open ; and whoever professes to prove the affirmative, ought not 

 to be charged with pretending to prove that the law of facility 

 is actually expressed by the function above mentioned. For 

 anjrthing that has yet been shown to the contrary, that function 

 may truly express our expectation of the unknown law, and the 

 true solution of the problem may be obtained by employing this 

 " provisional '^ law, as if it were a known or " definitive ^' law. 

 (See an analogous case discussed in the first paper, Phil. Mag. 



