ON PROBABILITY. 
1. IN considering any future event, we are generally unable to determine 
whether or not it will happen; yet, we can often conjecture the number of 
cases which are possible, and of these how many favour the production of 
the event in question. In our uncertainty, we say that there is a chance it 
will happen; and thus our idea of chance arises from our wanting data 
which might enable us to decide whether or not the event will take place. If] 
for instance, a bag contain one white and two black balls, it is impossible to 
decide whether or not a black ball will be drawn in one trial 5 but we know 
that there are three cases possible, of which two favour the appearance of a 
black ball and one the contrary, and of these, we have no reason to think 
one more probable than another. 
2. The operations of the mind are of two kinds ; the one consists in ac- 
quiring data, the other consists in making deductions from data previously 
acquired. Our data are only probable ; our deductions from these are also 
probable. The subject, therefore, of this treatise is intimately connected 
with every science, and, whether on account of its numerous and useful 
applications, or of the exact reasoning by which its principles are established, 
carries with it the highest degree of interest. In the sequel we shall explain 
the method of applying it to the calculation of life annuities, a few tables of 
which will be subjoined. 
3. To avoid circumlocution, those cases which embrace the production of 
a particular event are called the favourable cases, and those which do not, 
the unfavourable cases. It is usual to apply the word belief to the past, and 
po the word expectation to the future ; but the theory of probability is in all 
respects the same, whether it be applied to past or to future events. When 
we endeavour to discover whether an event Jot happen, we review the 
different cases which are possible. If the favourable cases are more numerous 
than the unfavourable, we Jos) that the event J take place, 
The words believe and expect, and those to which they correspond, are 
placed between brackets, in order to shew that the reasoning is the same in 
both cases. Let us suppose that a bag contains one black and two white 
balls: if I am asked whether a white ball will be drawn, or if, a ball being 
already drawn, but concealed from my view, I am asked whether a white 
ball has been drawn, it is clear that the judgment formed in both cases will 
be the same. I answer in both cases, that it is more probable that the ball 
which is drawn is white than black, yet if the ball be already drawn, but 
concealed from my view, the event is already determined and certain, We 
perceive, therefore, that the estimation of probability has no necessary re- 
ference to actual occurrence, but only to the means of judging which a given 
individual possesses. 
4. We have used the word probability before giving its definition, because 
its popular meaning has hitherto sufficed for our purpose. We now give its 
mathematical definition, which is this: the probability of any event is the 
ratio of the favourable cases to all the possible cases which, in our judgment, 
are similarly circumstanced with regard to their happening or failing. Thus, 
if a bag contain one white and two black balls, the probability of drawing 
a white ball is 1; the probability of throwing ace with a die at the first