Sec. 3.1 THE DETERMINATION OF PROBABILITIES 53 



Other classes of single events could be defined, of course, such as 

 the two classes: "sum ^ 5" and "sum > 5." From Table 3.11 it is 

 apparent that 10 of the 36 possible single events produce sums which 

 are less than or equal to 5, whereas the remaining twenty-six single 

 events yield sums which are greater than 5. Therefore, in this case, 

 N = 36, s — 2, n x = 10, and n 2 = 26; so that F(sum = 5) = n 2 /N 

 = 26/36 - 13/18, or = .72. 



Two useful facts are derivable from formula 3.11: 



(3.12) < P(E) < 1 because no m can be larger than N; and, 



(3.13) P(E) + P(not E) = 1 because m/N + (N - n t )/N 



= N/N = 1. 



Other laws follow from formula 3.11. Two of the more important 

 theorems will be proved and illustrated. Suppose that E x and E 2 

 denote two mutually exclusive classes of events; that is, single events 

 in classes Ex and E 2 cannot occur simultaneously on any one trial. 

 Suppose also that there are m and n 2 single events in classes E x and 

 E 2 , respectively. If a total of N single events is possible, the prob- 

 ability that an event in either class E x or class E 2 will occur on one 

 random trial is, by definition, 



(3.14) P(E X or E 2 ) = (m + n 2 )/N = m/N + n 2 /N 



= P(E X ) + P(E 2 ). 



The same reasoning and algebra are sufficient to show that for r 

 classes of events: E x , E 2 , . . . , E r with n x single events in class E t 

 (i = 1 to r) , the probability that some one of the mutually exclusive 

 events E x , E 2 , . . . , E r will occur on one random trial is given by 



(3.15) P(E u E 2 ,---,ovE r ) =P(E 1 )+P(E 2 )+--.+ P(E r ). 



This result is known as the Law of Total Probability for Mutually 

 Exclusive Events. 



To illustrate formula 3.15, suppose that a sack contains 10 green, 

 15 red, 5 white, and 20 purple marbles, all identical save for color. 

 What is the probability that a colored marble will be drawn on one 

 future random trial? Let E x stand for the drawing of any one of the 

 green marbles. There are 10 green marbles, and each is equally 

 likely to be drawn; hence there are 10 single events in the class E x . 

 Also, let E 2 represent the drawing of any red marble, E s stand for the 

 drawing of a white marble, and £J 4 equal the drawing of any purple 



