THE TllF.ORY 01' I'ROH.UULITIES 79 



falls on a first selector liaxing access to sub-group d, for example, 

 the second call still has the same chance of falling on a first selector 

 ha\ing access to sub-group Gi as on one having access to any one of the 

 other three sub-groups. In Case 3, however, the busy first selectors 

 tend to be distributed uniformly between the 4 sub-groups, so that 

 if any sub-group should have a preponderance of busy first selectors 

 the probability of its receiving another call is less than the probability 

 that one of the other sub-groups, with more idle first selectors, should 

 receive it. The full discussion of this case is reserved for the future. 



APPENDIX 

 Introduction to the Mathematical Theory of Probabilities 



If it is known that one of two events must occur in any trial or 

 instance, and that the first can occur in u ways and the second in v 

 ways, all of which are equally likeh- to happen, then the probability 

 that the first will happen is mathematically expressed by the fraction 



u + V 

 while the probability that the second will happen is 



u -\- V 

 Denote these probabilities by p and q respectively; then we have: 



^ = ^7T^' 5 = ir+-v ^ + 5 = 1. 



the last equation following from the first two, and being the mathe- 

 matical expression for the certainty that one of the two events must 

 happen. 



If the probabilities of two independent events are pi and p2 re- 

 spectively, the probability of their concurrence in any single instance 



is p\p2, and in general if pi, p2, pi, pn denote the probabilities 



of several independent events, and P the probability of their con- 

 currence, then 



P = pip2p3 pn. 



Consider, now, what may happen in n trials of an event, for which 

 the probability is p and against which the probability is q. The 



