PEOFESSOE BOOLE ON THE THEOET OE PEOBABILITIES. 
231 
and it only remains to eliminate X. Now from the above, 
— 2? ~ I ~ Q ~{~ 7* 1 
X X^-Oj X 2 5 
therefore 
^= 0 , ^> 0 , r> 0 , 
^—p + q + r—\ ^=p + q + r—\ ^—p + q + r—l 
!P> 2 ’ 2 ’ 2 
The last three conditions are reducible to the simpler form, 
1 , 1 , r>^+^— 1 . 
Such are the conditions of possible experience in the data. 
Suppose, for instance, it was affirmed as a result of medical statistics that, in two- 
fifths of a number of cases of disease of a certain character, two symptoms x and y were 
observed ; in two-thirds of the cases the symptoms y and z were observed ; and in four- 
fifths of the cases the symptoms z and x were observed ; so that, the number of cases 
observed being large, we might on a future outbreak of the disease consider the fractions 
•f, and -f as the probabilities of recurrence of the particular combinations of the 
symptoms x, y, and z observed. The above formulae would show that the evidence was 
contradictory. For, representing the respective fractions byp, q, and r, the condition 
q)>q-\-r—\ is not satisfied. {Edmhurgh Memoir.) 
In applying the above method to the ajqriori limitation of questions in the theory of 
probabilities, it will be necessary to represent the probability sought by a single letter 
w, and treat this as if it were one of the numerical data. The resolution of the event 
of which the probability is sought into alternatives belonging to the same scheme as 
those of the events in the data gives us a new equation, which must be combined with 
the equations involving jp, &c. The elimination of X, i^, &c. then determines not 
only the conditions of possible experience limiting q), q, r, but also the conditions which 
u must satisfy a ])riori, whatever method for its actual determination may be employed. 
Thus, if from the foregoing data it were required to determine the a qyriori limits of 
Prob. xyz, i. e. of the probability of the conjunction of the events x, y, z, we should 
have as the additional equation 
u=X, 
and therefore, after elimination of X, ///, v, 
u<p, ii^q., 
—p + q + r— 1 
2 ’ 
the conditions required. 
It will, however, in most of the following investigations suffice to consider the con- 
ditions of possible experience in the data alone, because it will be shown that when 
these are satisfied the corresponding conditions for the probability sought, when its value 
is determined by the method of the following section, will also be satisfied. 
