196 The Rule of Succession. [CHAP. vin. 



leaving the ground of past experience, but takes the conse 

 quences of this divergence as the express subject of its calcu 

 lation. It professes to give a general rule for the measure of 

 expectation that we should have of the reappearance of a 

 phenomenon that has been already observed any number of 

 times. This rule is generally stated somewhat as follows: 

 &quot;To find the chance of the recurrence of an event already 

 observed, divide the number of times the event has been 

 observed, increased by one, by the same number increased 

 by two.&quot; 



7. It will be instructive to point out the origin of 

 this rule ; if only to remind the reader of the necessity of 

 keeping mathematical formulae to their proper province, 

 and to show what astonishing conclusions are apt to 

 be accepted on the supposed warrant of mathematics. 

 Revert then to the example of Inverse Probability on p. 

 182. We saw that under certain assumptions, it would 

 follow that when a single white ball had been drawn 

 from a bag known to contain 10 balls which were white 

 or black, the chance could be determined that there was 

 only one white ball in it. Having done this we readily 

 calculate directly the chance that this white ball will 

 be drawn next time. Similarly we can reckon the chances 

 of there being two, three, &c. up to ten white balls in it, 

 and determine on each of these suppositions the chance 

 of a white ball being drawn next time. Adding these 

 together we have the answer to the question : a white 

 ball has been drawn once from a bag known to contain 

 ten balls, white or black ; what is the chance of a second 

 time drawing a white ball ? 



So far only arithmetic is required. For the next step we 

 need higher mathematics, and by its aid we solve this 

 problem : A white ball has been drawn m times from a 



