616 PROFESSOR BOOLE ON THE COMBINATION 
tion, it is the probability that if the observation be made under its actual circum- 
stances of care, personal fitness,’ instrumental accuracy, &c., it will be absolutely 
correct. Symbolically, it is the probability that if the event x take place, the 
event # will take place. The only mode of expressing this is by writing for the 
probability of # an arbitrary constant @, we have then 
Prob. =a, Prob. wiw=a, 6, % i ; (1) 
The events w and w are not, however, independent. If we can affirm that a 
given observation is correct, we can affirm that that observation has been made. 
Symbolically, the occurrence of the event 7 implies the occurrence of the event 
#. Expressing this proposition in the language of the calculus of Logic we have 
the equation. 
We=o . : , : ; (2) 
This forms a part of our data. It permits us to change also the form of one 
of the previous data, and instead of (1) to substitute 
Prob. v=a, Prob. w=a, ¢, : 5 : (3) 
In like manner, representing the arbitrary probability of the event y by a,, 
we have 
Prob. y=a, Prob. yu=a, ¢, ¢ - : (4) 
With the connecting condition 
A ie hematite ala ih ean ie lh (5) 
which would permit us to substitute for (4) the system 
Prob. y=a, Prob. v=a, ¢, 2 : é (6) 
Again, when it is known that the first observation is a correct one, the proba- 
bility that an indicator directed at random to the quadrant in which the star is 
situated will point below the star is p,. This, too, is a conditional probability. 
Symbolically, it is the probability that if the event s occur, the event z will 
occur. Hence, as the probability of the occurrence of 1 as a, ¢,, we have 
Prob. w z=a, ¢, Pp, - ‘ ‘ . F (7) 
In like manner we find 
Prob. vz=dy Cy Pp» : : , (8) 
Lastly, it is supposed that the values p and q are different. This involves the 
condition that the observations cannot both be correct. Whence we have the logi- 
cal equation. ™ 
wvu=0 : : J ; : (9) 
This completes the analysis of the logical elements involved in the data of the 
problem. We now proceed to analyse those involved in its queesitum or object 
proposed. 
That object is to determine the probability of the event z, when the occurrence 
