50 ELEMENTARY PROBABILITY Ch. 3 



the company. Whether or not such a person does live to receive a 

 particular payment must be regarded as a chance event and, there- 

 fore, requires some use of the theory of probability. Public opinion 

 polls regarding political matters, buyers' preferences, and foreign 

 affairs involve chance in the selection of the persons who are to be 

 interviewed. The reader should be able to think of many other 

 everyday events in which the theory of probability is involved. 



3.1 THE DETERMINATION OF PROBABILITIES 



Before a method is presented for determining the probability that 

 an event E will occur under specified conditions it is useful to dis- 

 tinguish between what will be called single events and classes of 

 events. For the purposes of this book this distinction can be made 

 by means of examples. Suppose that two dice are placed in a can, 

 shaken vigorously, and rolled out upon a flat, hard surface. Many 

 "events" can occur with each die, but just six usually are of interest: 

 a 1, 2, 3, 4, 5, or a 6 appears on the upper face of each die when it 

 stops rolling. How the dice were turned when they were thrown, 

 where on the surface they came to rest, or how many turns they 

 made while in motion are ordinarily of no interest. Moreover, it 

 would be at least impracticable, if not impossible, to relate those 

 phenomena to the number of dots on the upper face of a die. Hence 

 the six possible events which will be considered herein are the ap- 

 pearance of a 1, 2, 3, 4, 5, or a 6 on the upper face of each die. Since 

 these events cannot be further decomposed, we shall refer to them as 

 single events. If, with each die, these faces tend to appear with equal 

 relative frequencies over many trials, the dice are each said to be 

 unbiased. It is with single events occurring with equal relative fre- 

 quencies that we shall be primarily concerned in the subsequent dis- 

 cussion. If both dice are considered simultaneously and an event is 

 considered to consist of a number on one die and a number on the 

 other die, thirty-six single events are possible because any one face 

 on the first die can appear with any of the six faces on the other die. 

 Each possible pair of faces defines an observable event. 



If attention is turned to the sum of the numbers of dots appearing 

 on the upper faces of two dice which have been thrown simulta- 

 neously, any one of eleven different sums is possible. The different 

 possible sums define eleven classes of single events (occurring with 

 equal relative frequencies). For example, the class of events (com- 



