Sec. 3.1 THE DETERMINATION OF PROBABILITIES 51 



posed of such single events) , "sum = 7," contains the following single 



events : 



1 on die 1, 6 on die 2 



2 on die 1, 5 on die 2 



3 on die 1, 4 on die 2 



6 on die 1, 1 on die 2; 

 5 on die 1, 2 on die 2; 

 4 on die 1, 3 on die 2. 



The class, "sum = 2," includes but one single event because there is 

 but one way that it is possible to get a sum of 2. The class, "sum = 

 3," includes two single events: a 1 on die 1, a 2 on die 2; or a 2 on die 

 1, a 1 on die 2. The class, "sum = 4," includes three single events, 

 etc., until all thirty-six of the possible single events have been put 

 into one of the eleven classes of events. 



We could define other classes of events among the thirty-six single 

 events possible when two unbiased dice are tossed. For example, 

 we could have class 1 = "sum = 7" and class 2 = "sum is not = 7." 

 There are six single events in class 1 and thirty single events in class 2. 



The preceding discussion has brought out the fact that single events 

 and classes of events differ in one important respect. The single 

 events are expected to occur with equal relative frequencies over 

 many trials under the specified conditions, whereas the classes of 

 events consist of groupings of single events, and hence would be ex- 

 pected to occur with relative frequencies which depend upon the 

 numbers of single events in the classes. 



Upon the basis of the preceding discussion, a useful method for 

 determining probabilities can be devised for instances in which the 

 single events occur with unequal relative frequencies. Suppose that 

 under certain specified conditions any one of N possible single events 

 can occur and that they form an exhaustive set; that is, some one of 

 these single events must occur on any trial under the specified condi- 

 tions. Assume also that the single events are grouped into s non- 

 overlapping classes of events, with n\ in class 1, n 2 in class 2, . . . , 

 and with n s in class s. Then the probability that the single event 

 which actually does occur on one future trial will belong to class i 

 (i varies from 1 to s) is given by 



(3.11) P(Ei) = m/N. 



As an illustration of the use of formula 3.11 consider the dice prob- 

 lem discussed above in which thirty-six single events are possible. 

 Certain classes of events, the single events which each class includes, 

 and the probabilities associated with each class of events are given 

 in Table 3.11. 



