76 INDUCTION. 



$ 3. From these principles it is easy to deduce 

 the demonstration of that theorem of the doctrine of 

 probabilities, which is the foundation of its principal 

 application to judicial or other inquiries for ascertain- 

 ing the occurrence of a given event, or the reality of 



human actions) an equal chance whether I guess A or B; but it is 

 not, therefore, an equal chance whether A or B takes place. 



The fallacy has been stated thus. Suppose that either A or B 

 must happen: and let the chance that A will happen be x: as cer- 

 tainty is represented by 1, the chance that B will happen is \x. 

 Now, the chance that the event I guess will come to pass, is made 

 up of two chances: the chance that I shall guess A and that A 

 will happen, plus the chance that I shall guess B and that B will 

 happen. The chance that I shall guess A being \ ; the chance 

 that I shall guess A and that A will happen, is compounded of 

 ^ and x: it is therefore J x. The chance that I shall guess B 

 being also ^, the chance that I shall guess B and that B will 

 happen, is \ (1 #). But the sum of these two is : therefore the 

 chance that the event I guess will come to pass, is always an even 

 chance. But since it is an even chance that my guess will be right, 

 it is an even chance which of the two events will occur, whatever 

 may be their comparative frequency in nature. 



The whole of this reasoning is sound up to the last step, but 

 that step is a non sequitur. Before I have guessed, or until I have 

 made my guess known, it is an even chance that I guess right; but 

 when I have guessed, and guessed A, it is no longer an even chance 

 that I have guessed right ; otherwise there would be an even chance 

 in favour of the most improbable event. Let the question be, Is 

 Queen Victoria at this moment alive : and let me be required to guess 

 aye or no, without knowing about what, in order that I may be 

 equally likely to guess the one and the other. No one will say it 

 is an even chance which is true ; but it really is an even chance 

 whether my guess will be right. The chance of my guessing in 

 the negative and being right, is ^ of a very small chance, nay, 

 perhaps TO 0*000? but the chance of my guessing in the affirmative, 

 and being right, is \ of the remaining ^poolr '> so that the two 

 together are \. When, however, I have guessed, and told my 

 guess, the even chance which of the two I should guess is con- 

 verted into a certainty. If I have guessed aye, the chance that I 

 am right is yff^f : if no, it id only 



