4 ON THE FO UNDA TIONS OF 



A coin is to be thrown 100 times: there are 2 100 definite 

 sequences of heads and reverses, all equally possible if the coin 

 is fair. One only of these gives an unbroken series of 100 

 heads. A very large number give 50 heads and 50 reverses; 

 and Bernouilli's theorem shows that an absolute majority of the 

 2 100 possible sequences give the difference between the number of 

 heads and reverses less than 5. 



If we took 1000 throws, the absolute majority of the 2 100( 

 possible sequences give the difference less than 7, which is pro- 

 portionally smaller than 5. And so on. 



Now all this is not only true, but important. 



But it is not what we want. We want a reason for believing 

 that on a series of trials, an event tends to occur with frequency 

 proportional to its probability ; or in other words, that generally 

 speaking, a group of 100 or 1000 will afford an approximate 

 estimate of this probability. 



But, although a series of 100 heads can occur in one way 

 only, and one of 50 heads and 50 reverses in a great many, 

 there is not the shadow of a reason for saying that therefore the 

 former series is a rare and remarkable event, and the latter, 

 comparatively at least, an ordinary one. 



Non constatj but the single case producing 100 heads may 

 occur so much oftener than any case which produces 50 only, 

 that a series of 100 heads may be a very common occurrence, 

 and one of 50 heads and 50 reverses may be a curious anomaly. 



Increase the number of trials to 1000, or to 10,000. Pre- 

 cisely the same objection applies : namely, that in Bernouilli's 

 theorem, it is merely proved that one event is more probable 

 than another, i.e. by the definition can occur in more equally 

 possible ways, and that there is no ground whatever for saying, 

 it will therefore occur oftener, or that it is a more natural occur- 

 rence. On the contrary, the event shown to be improbable may 

 occur 10,000 times for once that the probable one is met with. 



To deny this, is to admit that if an event can take place in 

 more equally possible ways, it will take place more frequently. 

 But if this is admitted, Bernouilli's theorem is unnecessary. It 

 leaves the matter just where it was before, and introduces no 

 new element into the question. 



8. Thus, both by an appeal to consciousness, and by the 



