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PEOFESSOE BOOLE ON THE THEOET OE PEOBABILITIES. 
probabilities of the alternatives which it involves, we shall have a system of equations 
connecting X, (j!j, i>, See. with the probabilities supposed given. Again, 
A, jO;, y . . ., as probabilities, are subject to the conditions 
X>0, . . . &c., 
and, as alternatives mutually excluding each other, to the condition 
. . . = 1 , 
or the condition 
according as the alternatives in question together make up certainty or not. 
Thus we have a system consisting of equations and inequations from which X, (ju, v, Sec. 
must be eliminated. To effect this elimination we must determine as many of the quan- 
tities !«-, v . . . as possible from the equations, substitute their values in the inequations, 
and then eliminate the remainder of the quantities X, [Jb, v ... hy means of the theorem 
that if we have simultaneously 
X < X < Cli2^ ... X 
X ^2, X ^25 . . . X 
then we have the system of conditions of which the type is 
representing any one of the set ttj, and hj any one of the set 5,, . . . b^. 
Thus there are nm conditions in all. 
This method is illustrated in the following problem, in the expression and solution of 
'which it is to be noticed, that when in the Calculus of Logic an event is represented 
by cc, the event which consists in its not happening is denoted by 1 — x, or for brewty 
by X ; that when two events are represented by x and y, their concurrence is denoted 
by xy, the happening of the first without the second by xy, and so on. 
Problem. Given that the probability of the concurrence of the events x and y is jj, of 
the events y and 2 , and of the events z and r. Kequired the conditions to which 
j;, g-, and r must be subject in order that the above data may be consistent with a pos- 
sible experience. 
Eesolving the events ?/ 2 , xz into the possible alternations out of which they are 
formed, let us write 
Prob. Prob, Pxoh. x^z-=v, Prob. 
Then we have the equations 
X-\-iJj=i}, X+^= 2 ', X-\-v=r, 
together with the inequations 
X^O, /!A>0, v>0, ^>>0, 
From the equations we find 
[^=])-X, §=q-X, v=r^X, 
which, substituted in the inequations, give 
X > 0, q) — X >. 0, ^“"X^-O, j^“-X > Oj 
2X<1; 
