THE THEORY OF PROBABILITIES. 3 



to divest himself of the belief that the expected event will occur 

 more frequently than any other. 



For myself, after giving a painful degree of attention to the 

 point, I have been unable to sever the judgment that one event 

 is more likely to happen than another, or that it is to be 

 expected in preference to it, from the belief that on the long 

 run it will occur more frequently. 



5. It follows as a limiting case, that when we expect two 

 events equally, we believe they will recur equally on the long 

 run. In this belief we may of course be mistaken : if we are, 

 we are wrong in expecting the two events equally, and in think- 

 ing them equally possible. Conversely, if the events are truly 

 equally possible, they really will tend to recur equally on a 

 series of trials. But this proves the proposition placed at the 

 head of the section : for if any event can occur in a out of b 



equally possible ways, its probability is j-i and if all these 



b cases tend to recur equally on the long run, the event must 

 tend to occur a times out of b ; or in the ratio of its probability. 

 Which was to be proved. 



6. Let us now examine the mathematical demonstration 

 of this proposition. In entering upon it, we are supposed to 

 have no reason whatever to believe that equally possible events 

 tend to occur with equal frequency. 



It is well known that what is called Bernouilli's theorem, 

 relates to the comparative magnitudes of the several terms of 

 the binomial expansion. 



The general term of 



which is the probability that an event whose simple probability 

 is m will recur p times on Jc trials ; and hence the connexion 

 between the binomial expansion and the theory of probabilities. 



7. A particular example will suffice to illustrate what 

 seems to me to be the essential defect of the mathematical proof 

 of the proposition in question. 



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