234 THE PRINCIPLES OF SCIENCE. 



one head in two throws, but in our addition we have 

 involved the case in which two heads also appear. The 

 true result is ^ -f ^ x \ or |- , or the probability of head at 

 the first throw, added to the exclusive probability that if it 

 does not come at the first, it will come at the second. 

 Some of the greatest difficulties of the theory and the 

 subtlest errors arise from the confusion of exclusive and 

 unexclusive alternatives. I may remind the reader that 

 the possibility of unexclusive alternatives was a point 

 previously discussed (p. 81), and to the reasons then given 

 for considering alternation as logically unexclusive, may be 

 added the existence of these difficulties in the theory of 

 probability. The expression 



Headfirst throw or head second throw 

 ought to be interpreted in our logical system as including 

 both cases at once, and so it is in practice. 



Employment of the Logical Abecedarium in questions of 



Probability. 



When the probabilities of certain events are given, and 

 it is required to deduce the probabilities of compound 

 events, the Logical Abecedarium may give assistance, pro- 

 vided that there are no special logical conditions and all 

 the combinations are possible. Thus, if there be three 

 events A, B, C, of which the probabilities are a, (3, y, then 

 the negatives of those events, expressing the absence 

 of the events, will have the probabilities i a, i /3, iy. 

 We have only to insert these values for the letters of the 

 combinations and multiply, and we obtain the probability 

 of each combination. Thus the probability of ABC is 

 a/3 7 ; of Kbc, a(i -/3)(l - 7). 



We can now clearly distinguish between the probabilities 

 of exclusive and unexclusive events. Thus if A and B 

 are events which may happen together like rain and high 



