2 Mr. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES. 



another will occur more frequently; or a case in which he may be able to divest himself of 

 the belief that the expected event will occur more frequently than any other. 



For myself, after o-iving 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 behef 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 wron^ in expecting the two events equally, and in thinking 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 h equally possible ways, its probability is — : and if all these 6 cases 



tend to recur equally on the long run, the event must tend to occur a times out of 6 ; or in the 

 ratio of its prpbability. Which was to be proved. 



(C.) 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 mag- 

 nitudes of the several terms of the binomial expansion. 



\k\ 

 The general terra of !>» + (1 - "»)}', is ^ ^ , . r ™'' (l - »n)'"^, which is the probability 



' [p][&-p] 



that an event whose simple probability is m will recur p times on k trials ; and hence the 

 connexion between the binomial ex2iansion 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. 



A coin is to be thrown 100 times : there are s'"" 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'°° possible sequences give the difitrence between the number of 

 lieads and reverses less than 5. 



If we took 1000 throws, the absolute majority of the s"""* possible sequences give the 

 difference less than 7, which is proportionally 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, althougli 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. 



No7i constat, 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 Ije 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. Precisely 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 

 occurrence. 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, 



