11/5 THR FULLY -JOINED SYSTEM 



Suppose N events each have a probability p of success, and 

 the probabilities are independent. An example would occur if N 

 wheels bore letters A and B on the rim, with A's occupying the 

 fraction p of the circumference and 2?'s the remainder. All are 

 spun and allowed to come to rest; those that stop at an A count 

 as successes. Let us compare three ways of compounding these 

 minor successes to a Grand Success,* which, we assume, occurs 

 only when every wheel is stopped at an A. 



Case 1: All N wheels are spun; if all show an A, Success is 

 recorded and the trials ended; otherwise all are spun again, and 

 so on till ' all A's ' comes up at one spin. 



Case 2: The first wheel is spun; if it stops at an A it is left 

 there; otherwise it is spun again. When it eventually stops at an 

 A the second wheel is spun similarly; and so on down the line of 

 N wheels, one at a time, till all show^'s. 



Case 3: All N wheels are spun; those that show an A are left 

 to continue showing it, and those that show a B are spun again. 

 When further A's occur they also are left alone. So the number 

 spun gets fewer and fewer, until all are at A's. 



Regard each spin (regardless of the number of wheels turned) 

 as one trial. We now ask, how many trials, on the average, will 

 the three cases require ? 



Case 1 will require ( - ) N , as in S. 11/2. Case 2 will require, on 



the average, l/'p for the first wheel, then \/p for the second, and 

 so on; and thus N/p for them all. Case 3 is difficult to calculate 

 but will be the average of the longest of a sample of N drawn from 

 the distribution of length of run for one wheel ; it will be somewhat 

 larger than I /p. 



The calculations are of interest not for their quantitative exact- 

 ness but because when N gets large they tend to widely differing 

 values. Suppose, for instance, that p is \, that spins occur at 

 one a second, and that N is 1,000. Then if T v T 2 , and T 3 are 

 the average times to reach Success in Cases 1, 2, and 3 respectively, 

 Ti = 2 iooo seconds, 



T 2 = — — — seconds, 



Li 



T 3 = rather more than \ second. 



* As we shall have to consider several compoundings of minor events to 

 major events, I shall use the convention of I. to C, S. 13/8, and distinguish 

 them respectively by a lower case, or a capital, initial. 



151 



