600 PROFESSOR BOOLE ON THE COMBINATION 
6. I define the mathematical probability of an event as the ratio which the 
number of distinct cases or hypotheses favourable to that event bears to the 
whole number of distinct cases possible, supposing that to none of those cases 
the mind is entitled to give any preference over any other. Fundamentally, this 
definition agrees with that of Lartace. “La théorie des hazards consiste,” he 
remarks, “4 réduire tous les évenements du méme genre a un certain nombre de 
cas également possibles c’est 4 dire tels que nous soyons également indécis sur leur 
existence et 4 déterminer le nombre de cas favorables 4 ’évenement dont on 
cherche la probabilité. Le rapport de ce nombre a celui de tous les cas possibles 
est la mesure de cette probabilité.”—Essai Philosophique sur les Probabilités. 
It is implied in this definition, that probability is relative to our actual state 
of information, and varies with that information. Of this principle LapLace 
gives the following illustration :—* Let there be three urns, A, B, C, of which we 
are only informed that one contains black and the other white balls; then, a ball 
being drawn from C, required the probability that the ball is black. As we are 
ignorant which of the urns contains black balls, so that we have no reason to 
suppose it to be the urn C rather than the urn A or the urn B, these three hy- 
potheses will appear equally worthy of credit, but as the first of the three hypo- 
theses alone is favourable to the drawing of a black ball from C, the probability 
of that event is 4. Suppose now that, in addition to the previous data, it is 
known that the urn A contains only white balls, then our state of indecision has 
reference only to the urns B and C, and the probability that a ball drawn from 
C will be black is 3. Lastly, if we are assured that both A and B contain white balls 
only, the probability that a black ball will issue from C rises into certitude.’— 
Essai Philosophique sur les Probabilités, p. 9—(Phil. Mag., p. 433.) Our estimate 
of the probability of an event varies not absolutely with the circumstances which 
actually affect its occurrence, but with our knowledge of those circumstances. 
7. When the probabilities of simple events constitute our only data, we can, 
by virtue of the above definition, determine the probability of any logical combi- 
nation of those events, and this either, 1s¢, absolutely; or, 2d/y, conditionally. 
The reason why we can, in this case, more immediately apply the definition is, 
that not only is no connection expressed among the events whose probabilities are 
given, but none is implied, nor is any restraint imposed upon their possible com- 
binations. This, as we shall see, is not the case when the data are the probabi- 
lities of compound events. 
As an example, let us suppose that the probability of the conjunction of two 
events, « andy, is required, the data being simply that the probability of the 
event z is p, and that of the event y is g. Or, to express the problem in a form 
which we shall hereafter generally employ: 
Given Prob.v=p, Prob. y=q, 
Required Prob. xy. 
