36G Prof. Karl Pearson on the Influence of 



expected, the result anticipated will be mp individuals of the 

 given character with a probable error of *67449a/^P^. This 

 result is often given without any regard to the relative mag- 

 nitudes of m and n, or again of p and q. Further, tables of 

 the probability integral are applied to determine the proba- 

 bility of deviations, although many of the users would be 

 much puzzled to cite where the justification for using the 

 normal curve at all is to be found. 



It seems therefore desirable to reinvestigate the whole 

 subject. I start as most mathematical writers have done, 

 with " the equal distribution of ignorance," or I assume the 

 truth of Bayes' Theorem. I hold this theorem not as rigidly 

 demonstrated, but I think w 7 ith Eclgeworth that the hypothesis 

 of the equal distribution of ignorance is, within the limits of 

 practical life, justified by our experience of statistical ratios, 

 which, a priori are unknown, i. e. such ratios do not tend to 

 cluster markedly round any particular value. ci Chances " 

 lie between and 1, but our experience does not indicate any 

 tendency of actual chances to cluster round any particular 

 value in this range. The ultimate basis of the theory of 

 statistics is thus not mathematical but observational. Those 

 who do not accept the hypothesis of the equal distribution 

 of ignorance and its justiti cation in observation, are com- 

 pelled to produce definite evidence of the clustering of 

 •chances, or to drop all application of past experience to the 

 judgment of probable future statistical ratios. It is perfectly 

 easy to form new statistical algebras with other clustering of 

 chances. We can form other geometries with varied axioms 

 as to parallels. But as the usual axiom of parallels leads to 

 results in close approximation to experience, so the " equal 

 distribution of ignorance" leads us to results which accord 

 reasonably wdth actual observation. As to the second alter- 

 native, even those who criticise Bayes' Theorem apply at every 

 turn in practical life their past experience of statistical ratios 

 to actual conduct. 



With this prelude, which appears needful in the present 

 state of opinion as to the basis of probability, I turn to the 

 actual problem before me. 



(2) Let the chance of a given event occurring be supposed 

 to lie between x and x + 8x, then if on n = p + q trials an 

 event has been observed to occur p times and fail q times, 

 the probability that the true chance lies between x and# + &r 

 is, on the equal distribution of our ignorance. 



p _ x P{l-x)«dx 



x -£*{i-*y* w 



This is Bayes' Theorem. 



