DESIGN FOR A BRAIN 11/6 



When these values are converted to ordinary quantities, T x is 

 about 10 293 years, T 2 is about 8 minutes, and T 3 is a few seconds. 

 Thus, while getting Success under the rules of Case 1 (all simul- 

 taneously) is practically impossible, getting it under Cases 2 and 3 

 is easy. 



The final conclusion — that Case 1 is very different from Cases 

 2 and 3 — does not depend closely on the particular values of p 

 ahd N. It illustrates the general fact that the exponential 

 function (Case 1) tends to become large at an altogether faster 

 rate than the linear. If the reader likes to try other numbers 

 he is likely to arrive at results showing much the same features. 



11/6. Comparison of the three Cases soon shows why Cases 2 

 and 3 can arrive at Success so much sooner than Case 1 : they can 

 benefit by partial successes, which 1 cannot. Suppose, for 

 instance, that, under Case 1, a spin gave 999 A's and 1 B. This is 

 very near complete Success; yet it counts for nothing, and all the 

 A's have to be thrown back into the melting-pot. In Case 3, 

 however, only one wheel would remain to be spun; while Case 2 

 would perhaps get a good run of A's at the left-hand end and could 

 thus benefit somewhat. 



The examples thus show the great, the very great, reduction 

 in time taken that occurs when the final Success can be reached 

 by stages, in which partial successes can be conserved and 

 accumulated. 



11/7. It is difficult to find a real example which shows in one 

 system the three ways of progression to Success, for few systems 

 are constructed so flexibly. It is, however, possible to construct, 

 by the theory of probability, examples which show the differences 

 referred to. Thus suppose that, as the traffic passes, we note the 

 final digit on each car's number-plate, and decide that we want 

 to see cars go past with the final digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, in 

 that order. If we insist that the ten cars shall pass consecutively, 

 as in Case 1, then on the average we shall have to wait till about 

 10,000,000,000 cars have passed: for practical purposes such an 

 event is impossible. But if we allow success to be achieved by 

 first finding a ' ', then finding a ' 1 ', and so on until a ' 9 ' is 

 seen, as in Case 2, then the number of cars which must pass will 

 be about fifty, and this number makes ' Success ' easily achievable. 



152 



