2 
The Fuiidaiuen tal Problem of Practical Statistics 
the fundamental principle on which Laplace's more complete solution is based may 
be extricated from Bayes' much more involved reasoning. None of the early writers 
on this topic — all approaching the subject from the mathematical theory of games 
of chance — seem to have had the least inkling of the enormous extension of their 
ideas, which would result in recent times from the application of the theory of 
random sampling to every phase of our knowledge and experience — economic, 
social, medical, and anthropological — and to all branches of observation whether 
astronomical, physical or psychical. Hence they must not be too severely handled 
if their "bag of balls" hypotheses seem too Himsy a structure for modern 
statistical theory. 
Bayes in his original paper supposes balls rolled on a table and equally likely 
to halt anywhere on the breadth of the table. Condorcet and Laplace only generalise 
this in appearance by supposing all probabilities equally likely before the event. 
It is the same hypothesis expressed in more general words. Each repetition is 
supposed to be independent and the elementary algebra view of compound prob- 
ability is accepted without hesitation. Thus we have two hypotheses to deal with : 
(1) the hypothesis that a priori we ought to distribute our ignorance of the 
chance of a marked individual occiu-ring equally, 
(ii) the hypothesis that earlier occurrences do not modify the chance of 
later trials. 
(2) The chief criticism of the theory of inverse probabilities has been based on 
the want of generality in the first hypothesis — what Boole has termed " the equal 
distribution of our ignorance " {Laws of Th.ougJit, p. 370, 1854) or the assigning to 
the appearance of a factor whose real probability is unknown to us all degrees of 
probability and then making these degrees all equally likely to occur. This is not 
the sole, but I think the chief feature of Boole's attack on the theory of inverse 
probabilities. Why should we, he asks, distribute our ignorance equally ? Other 
sorts of distributions may occur, and we know do occur in chance problems. Why 
this assumption in the particular case ? Dr Venn's strongly unfavourable criticism 
of inverse probabilities seems also based on objection to the principle of equal 
distribution of ignorance*. Strangely enough it seems to me, he is willing to admit 
the hypothesis when it is applied to a comparison of the probable contents of two 
bags of balls, while refusing to consider it valid in the case of a single bag. Thus 
he considers the problem : " of 10 cases treated by Lister's method seven did well 
and three suffered from blood-poisoning: of 14 treated with ordinary dressings nine 
did well and five had blood poisoning, what are the odds that the success of Lister's 
method was due to chance " ? (p. 187) and remarks that the " bag of balls " method — 
i.e. the assumption of the equal prevalence of the different possible kinds of bag — 
seems " to be the only reasonable way of treating the problem, if it is to be 
considered capable of numerical solution at all." But I fail to see why the applica- 
tion of the hypothesis of the equal distribution of ignorance to Lister's method and 
* The Logic of Chance, p. 182 et seq. 
