THE THEORY OP PKOBABI I.]TI ES 75 



or choices. The reader unfamiliar with automatic systems may 

 consider a 10 level selector as one from which calls may be sent in 10 

 difTcrent directions. Assume that each level is equipped with 8 

 trunks to second selectors. The 591 line switches are multipled 

 together so that one group of 35 first selectors must handle the calls 

 originating from these 591 lines. The 35 first selectors are multi- 

 pled together so that one group of 8 second selectors must handle 

 the calls originating from the 591 lines for a particular level. It is 

 assumed that the 591 calls are distributed at random with reference 

 to the 10 levels of the first selectors. 



The probability that .Y should fail to obtain immediately a first 

 selector can be determined as in the first problem, but now let us 

 determine what is the probability that subscriber X (having obtained 

 immediately a first selector) fails to obtain immediately one of the 

 8 trunks of a particular one of the 10 levels on the first selectors. 



For subscriber Y to interfere with X it is necessary that Y originate 

 his call in the two minutes preceding the instant at which X calls 

 and also that Y call for the particular one of the 10 lev^els in which 

 X is interested. 



The probability of Y fulfilling the first condition is equal to the 

 probability of throwing the ace with a 30 face die. The probability of 

 Y fulfilling the second condition is equal to the probability of throw- 

 ing the ace with a 10 face die. 



The question may then be stated in the form of a dice problem as 

 follows: 591 throws are made with a 30 face die giving C aces. C 

 throws are made with a 10 face die giving D aces, and the question 

 is the probability that D is not less than 8. Assuming no restric- 

 tion ^ on the value of Cthis probability is the same as that of throwing 

 at least 8 aces in 591 throws with a die having (30) (10) = 300 faces. 



The average number of aces to be expected is (591/300) = 1.97 and 

 with this average the tables tell us that once in a thousand times 

 we may expect at least 8 aces. 



' Since it is assumed that X obtained a first selector it follows that in the 2 minutes 

 preceding the instant when X called the number of calls must have been less than 

 the number of first selectors and we should, therefore, not count the throws giving 

 values of C which are not less than the total number of first selectors. This res- 

 triction becomes of practical importance only where a large proportion of the calls 

 from the first selectors go to one level. To take an extreme case, assume that all 

 the calls went to one level, and that therefore each 10 first selectors would require 

 10 second selectors to handle the traffic. Placing no restriction on the value of C, 

 since C exceeds the number of first selectors occasionally, we would get the result 

 that 10 second selectors were not enough to handle all the calls from 10 first selectors, 

 which is of course absurd. Where, however, the values of C e.xceeding the number 

 of first selectors are assumed to be distributed over all 10 levels of the first selectors 

 their effect on the number of second selectors is negligible. 



