24 ON PROBABILITY. 
which serves to show how much more rapidly the probability increases than 
the number of trials ; for after the first 15,700 trials, the addition of 3500 new 
trials increases the odds tenfold, and 8500 more increases them tenfold 
again. 
38. This highly important theorem is due to James Bernoulli, a cele- 
brated mathematician of the last century, whose name it bears. 
39. We have supposed known the number of favourable and unfavourable 
cases similarly circumstanced in the problems we have hitherto considered. 
Thus, in the problems relating to dice, we took for granted the form of the 
dice, and also their homogeneity. 
40. However nearly any die may fulfil these given conditions, it will not do 
so strictly ; and if we investigate the probability of any throw upon the prin- 
ciples hitherto developed, we obtain a result approximately correct, and of 
which the error depends on the inaccuracy of our hypothesis. The know- 
ledge that a defect of homogeneity is possible, renders the return of the same 
face in several repeated trials more probable than it would be, if the die were 
known to be homogeneous. In tossing up a shilling, the probability of its 
falling heads, or the reverse, twice successively, is rather greater than 1. 
41. If such considerations apply to these very simple questions, it will . 
readily be seen how difficult it is to estimate mathematically those probabili- bu 
ties which depend on more complicated circumstances ; as the probability of 
an individual living a given number of years, or the probability of the truth of 2 
any assertion. Truths of definition are the only certain propositions. We ; 
shall not stop to inquire whether any limit separates truths of definition 
from propositions which rest upon experience; the distinction however may 
be admitted. 
42. It is impossible to suppose that “a part is greater than the whole” 
without involving a contradiction to the sense in which the words forming 
this sentence are understood, so that the truth of the proposition, that “a 
part is less than the whole” results from the very definition of the words 
which compose this sentence. If on the other hand we consider the propo- 
sition that the sun will rise to-morrow ; the number of times we believe the ) 
sun to have risen daily without interruption induces ns to believe that the sun 
will rise again. Most of our opinions arise from our experience of the past, 
and rest upon probabilities of this kind. 
43. In order to obtain mathematical solutions of problems similar to these 
we must revert to games of chance. Any problem in chances may be repre- 
sented by throws with dice of different forms, or by drawings from bags con- 
taining balls of different colours. Nor is it any objection to the results we 
obtain by this means, that no dice can be formed which exactly fulfil the con- 
ditions we suppose them to do, any more than itis to the theorems in Euclid 
that the lines which compose the diagrams are not mathematically straight, 
44. When our knowledge of the number of cases similarly circumstanced is 
imperfect, the probability of an event is still deduced upon the same principles 
as those hitherto developed. We have recourse to hypotheses, and having 
estimated their probability, the probability of any future event which depends 
on them, is easily deduced. 
45. The probability of each hypothesis by definition is the number of cases 
which favour this hypothesis divided by the whole number of cases pos- 
sible. 
Ex. 16. Let us suppose a bag to contain three balls, and that we are uncer- 
tain whether of these balls two are black and one is white, or one is black and 
two are white, and that a white ball has been drawn, 
Let us call these balls 1, 2, 3, and let us also suppose that the uncertainty