x.] THE THEORY OF PROBABILITY. 205 



would lead us to a probability of i| or of any greater 

 number, results which could have no meaning whatever. 

 The probability we wish to calculate is that of one head in 

 two throws, but in our addition we have included the case 

 in which two heads appear. The true result is + x 

 or f , 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. The greatest difficulties 

 of the theory 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. 68), 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 erroneous result explained above really 

 arose from overlooking the fact that the expression " head 

 first throw or head second throw " might include the case 

 of head at both throws. 



The Logical Alphabet in questions of Probability. 



When the probabilities of certain simple events are 

 given, and it is required to deduce the probabilities of 

 compound events, the Logical Alphabet may give assist- 

 ance, provided that there are no special logical conditions 

 so that all the combinations are possible. Thus, if there be 

 three events, A, B, C, of which the probabilities are, a, ft, 

 7, then the negatives of those events, expressing the absence 

 of the events, will have the probabilities I a, I fi, 17. 

 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 

 o#y; of A6c, a(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 

 tide, or an enrthquake and a storm, the probability of A or 

 B happening is not the sum of their separate probabilities. 

 For by the Laws of Thought we develop A -|- B into 

 AB'|- Afe'l-aB, and substituting a and ft, the probabili- 

 ties of A and B respectively, we obtain a./3+a.(i /S)+ 

 (l a)./S or a+fi a.p. But if events are incompossible 



