On certain Questions relating to the Theory of Probabilities. 459 



represents the quantity of our belief beforehand that the grains 

 will be distributed in the proposed way. As a numerical ex- 

 ample, suppose six grains thrown into a box divided into three 

 compai'tments ; the seven possible modes of distribution are 

 6,0,0; 1,5,0; 2,4,0; 3,3,0; 1,1,4; 1,2,3; 2,2,2; and 

 their respective probabilities are fractions whose common deno- 

 minator is 243, and numerators 1, 12, 30, 20, 30, 120, 30. In 

 the general case, the probability of equal distribution, or of each 



^ ^ . . . . 1.2.3. ..n 



compartment contammg r grams, is n i-i o ^ — Am ' 



3. Now it would of course be a gross fallacy in principle, 

 though it may often happen to produce small numerical errors, 

 to confound these a priori probabilities with the a posteriori pro- 

 babilities which are the subjects of investigation in cases where 

 a particular distribution is known to exist, or is asserted to have 

 happened. And nothing can be objected to Professor Forbes^s 

 remarks on this head, except that he appears to think that 

 mathematicians are in the habit of conmiitting the fallacy in 

 question, and even to doubt the possibility of treating such pro- 

 blems mathematically at all. On this part of the subject so 

 much was said in the former paper that little need be here added. 

 There are two questions, however, which deserve further discus- 

 sion, especially as one of them involves an apparent difficulty. 



4. The first question is, what is the account of our surprise 

 when we know, or believe, that a symmetrical event (such as the 

 equal distribution of the grains of rice) has happened accidentally ? 

 It is certainly not accounted for by a comparison of the u priori 

 probabilities of the possible symmetrical and unsymmetrical 

 events ; because, although in the case supposed an equal distri- 

 bution is really less probable a jiriori than many other possible 

 distributions, it is easy to conceive other cases in which all the 

 a priori probabilities are equal, but in which our sm'prise at a 

 symmetrical event would 1)6 quite as great. For instance, suppose 

 100 counters marked 1, 2, 3, &c. thrown into a box containing 

 100 compartments similarly numbered. The probability that 

 each counter will fall into the compartment marked with the 

 corresponding number, is beforehand exactly the same as the 

 probability of any other determinate event. Yet if we had reason 

 to believe that such a distribution, or any other following a 

 simple law, had really happened, we should be very nmch sui-- 

 prised ; whereas any irregular distribution would excite no sur- 

 prise at all. 



5. The most obvious answer to this question at first sight 

 wfjuid be, that we mentally divide all the possible events into 

 two |)arcels ; one containing all the reyulur or symmetrical events, 

 and the other all the irregular events ; and that the former parcel 



212 



