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 w will take place. The only mode of expressing this is by writing for the 

 probability of x an arbitrary constant a x we have then 



Prob. x = a l Prob. oc tv = a x c l ... (1) 



The events x 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 w implies the occurrence of the event 

 x. Expressing this proposition in the language of the calculus of Logic we have 

 the equation. 



w cc = 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. x = a l Prob. w — a x c x . . . (3) 



In like manner, representing the arbitrary probability of the event y by a „ 

 we have 



Prob. y = a 2 Prob. yv=a 2 c 2 ... (4) 



With the connecting condition 



vy=Q . . . (5) 



which would permit us to substitute for (4) the system 



Prob. y = a 2 Prob. v—a 2 c 2 ... (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 v . This, too, is a conditional probability. 

 Symbolically, it is the probability that if the event w occur, the event z will 

 occur. Hence, as the probability of the occurrence of w as a x c lf we have 



Prob. w z—a x c Y p x . . . . . (7) 

 In like manner we find 



Prob. vz = a 2 c 2 p 2 .... (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. 



w v=0 (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 qusesitum or object 

 proposed. 



That object is to determine the probability of the event z, when the occurrence 



