9/3 AN INTRODUCTION TO CYBERNETICS 



estimate. He might go on to collect third and fourth estimates. 

 If these several estimates proved seriously discrepant he would say 

 that rain at Manchester had no constant probability. If however 

 they agreed, he could, if he pleased, say that the fraction at which 

 they agreed was the constant probability. Thus an event, in a very 

 long sequence, has a "constant" probability of occurring at each 

 step if every long portion of the sequence shows it occurring with 

 about the same relative frequency. 



These words can be stated more accurately in mathematical terms. 

 What is important here is that throughout this book any phrases 

 about "probability" have objective meanings whose vahdity can 

 be checked by experiment. They do not depend on any subjective 

 estimate. 



Ex. 1 : Take the five playing cards Ace, 2, 3, 4, 5. Shuffle them, and lay them 

 in a row to replace the asterisks in the transformation T: 



rj., , Ace 2 3 4 5 



Is the particular transformation so obtained determinate or not? (Hint: 

 Is it single-valued or not?) 

 Ex. 2: What rule must hold over the numbers that appear in each column of a 

 matrix of transition probabilities? 



Ex. 3: Does any rule like that of Ex. 2 hold over the numbers in each row? 



Ex. 4: If the transformation defined in this section starts at 4 and goes on for 

 10 steps, how many trajectories occur in the set so defined? 



Ex. 5: How does the kinematic graph of the stochastic transformation differ 

 from that of the determinate ? 



9/3. The stochastic transformation is simply an extension of the 

 determinate (or single valued). Thus, suppose the matrix of transi- 

 tion probabilities of a three-state system were : 



The change, from the first matrix to the second, though small (and 

 could be made as small as we please) has taken the system from the 

 obviously stochastic type to that with the single-valued trans- 

 formation : 



ABC 



^ B A C 



164 



